(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 97463, 2652]*) (*NotebookOutlinePosition[ 98134, 2676]*) (* CellTagsIndexPosition[ 98090, 2672]*) (*WindowFrame->Normal*) Notebook[{ Cell["\<\ Examples for Section 8.8 Power Series\ \>", "Subsection"], Cell[CellGroupData[{ Cell["p. 671", "Subsection"], Cell[CellGroupData[{ Cell["3.", "Subsubsection"], Cell[TextData[{ "Find the radius and interval of convergence of the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\) \((4 x + 1)\)\^n\)]], ".\n\nSince this is a geometric series, we know it will converge absolutely \ if ", Cell[BoxData[ \(TraditionalForm\`\( | 4 x + 1 | \( < 1\)\)\)]], " (to the sum ", Cell[BoxData[ \(TraditionalForm\`1\/\(1 + \((4 x + 1)\)\)\)]], ") and diverge if ", Cell[BoxData[ \(TraditionalForm\`\( | 4 x + 1 | \( \[GreaterEqual] 1\)\)\)]], ". (The Root Test is based on the geometetric series and gives the same \ conclusion.) The inequality ", Cell[BoxData[ \(TraditionalForm\`\( | 4 x + 1 | \( < 1\)\)\)]], " is equivalent to ", Cell[BoxData[ \(TraditionalForm\`\( | x - \((\(-\(1\/4\)\))\) | \( < 1\/4\)\)\)]], " which shows that the center is ", Cell[BoxData[ \(TraditionalForm\`\(-\(1\/4\)\)\)]], "and the radius of convergence is ", Cell[BoxData[ \(TraditionalForm\`1\/4\)]], ". The corresponding interval is ", Cell[BoxData[ \(TraditionalForm\`\(-\(1\/2\)\) < x\ < 0\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p10\ = \ \[Sum]\+\(n = 0\)\%10\(\((\(-1\))\)\^n\) \((4 x + 1)\)\^n\)], "Input"], Cell[BoxData[ \(\(-4\)\ x + \((1 + 4\ x)\)\^2 - \((1 + 4\ x)\)\^3 + \((1 + 4\ x)\)\^4 - \((1 + 4\ x)\)\^5 + \((1 + 4\ x)\)\^6 - \((1 + 4\ x)\)\^7 + \((1 + 4\ x)\)\^8 - \((1 + 4\ x)\)\^9 + \((1 + 4\ x)\)\^10\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{1\/\(2 + 4 x\), p10}, {x, \(-1\)/2, 0}, PlotRange \[Rule] {0, 10}, PlotStyle \[Rule] {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.97619 1.90476 0 0.0618034 [ [.02381 -0.0125 -12 -9 ] [.02381 -0.0125 12 0 ] [.21429 -0.0125 -12 -9 ] [.21429 -0.0125 12 0 ] [.40476 -0.0125 -12 -9 ] [.40476 -0.0125 12 0 ] [.59524 -0.0125 -12 -9 ] [.59524 -0.0125 12 0 ] [.78571 -0.0125 -12 -9 ] [.78571 -0.0125 12 0 ] [.96369 .12361 -6 -4.5 ] [.96369 .12361 0 4.5 ] [.96369 .24721 -6 -4.5 ] [.96369 .24721 0 4.5 ] [.96369 .37082 -6 -4.5 ] [.96369 .37082 0 4.5 ] [.96369 .49443 -6 -4.5 ] [.96369 .49443 0 4.5 ] [.96369 .61803 -12 -4.5 ] [.96369 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 0 m .02381 .00625 L s [(-0.5)] .02381 -0.0125 0 1 Mshowa .21429 0 m .21429 .00625 L s [(-0.4)] .21429 -0.0125 0 1 Mshowa .40476 0 m .40476 .00625 L s [(-0.3)] .40476 -0.0125 0 1 Mshowa .59524 0 m .59524 .00625 L s [(-0.2)] .59524 -0.0125 0 1 Mshowa .78571 0 m .78571 .00625 L s [(-0.1)] .78571 -0.0125 0 1 Mshowa .125 Mabswid .0619 0 m .0619 .00375 L s .1 0 m .1 .00375 L s .1381 0 m .1381 .00375 L s .17619 0 m .17619 .00375 L s .25238 0 m .25238 .00375 L s .29048 0 m .29048 .00375 L s .32857 0 m .32857 .00375 L s .36667 0 m .36667 .00375 L s .44286 0 m .44286 .00375 L s .48095 0 m .48095 .00375 L s .51905 0 m .51905 .00375 L s .55714 0 m .55714 .00375 L s .63333 0 m .63333 .00375 L s .67143 0 m .67143 .00375 L s .70952 0 m .70952 .00375 L s .74762 0 m .74762 .00375 L s .82381 0 m .82381 .00375 L s .8619 0 m .8619 .00375 L s .9 0 m .9 .00375 L s .9381 0 m .9381 .00375 L s .25 Mabswid 0 0 m 1 0 L s .97619 .12361 m .98244 .12361 L s [(2)] .96369 .12361 1 0 Mshowa .97619 .24721 m .98244 .24721 L s [(4)] .96369 .24721 1 0 Mshowa .97619 .37082 m .98244 .37082 L s [(6)] .96369 .37082 1 0 Mshowa .97619 .49443 m .98244 .49443 L s [(8)] .96369 .49443 1 0 Mshowa .97619 .61803 m .98244 .61803 L s [(10)] .96369 .61803 1 0 Mshowa .125 Mabswid .97619 .0309 m .97994 .0309 L s .97619 .0618 m .97994 .0618 L s .97619 .09271 m .97994 .09271 L s .97619 .15451 m .97994 .15451 L s .97619 .18541 m .97994 .18541 L s .97619 .21631 m .97994 .21631 L s .97619 .27812 m .97994 .27812 L s .97619 .30902 m .97994 .30902 L s .97619 .33992 m .97994 .33992 L s .97619 .40172 m .97994 .40172 L s .97619 .43262 m .97994 .43262 L s .97619 .46353 m .97994 .46353 L s .97619 .52533 m .97994 .52533 L s .97619 .55623 m .97994 .55623 L s .97619 .58713 m .97994 .58713 L s .25 Mabswid .97619 0 m .97619 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .5 Mabswid .07155 .61803 m .07299 .59846 L .08269 .49987 L .09312 .4246 L .10458 .36437 L .11478 .3235 L .12409 .29348 L .145 .24283 L .16409 .2098 L .18485 .18275 L .20462 .16276 L .22563 .14582 L .26338 .12284 L .30359 .10519 L .34227 .09241 L .38341 .08184 L .42304 .07372 L .46115 .06729 L .50171 .06158 L .54075 .05693 L .58225 .0527 L .62223 .04918 L .6607 .04621 L .70162 .04342 L .74102 .04103 L .77891 .03898 L .81925 .037 L .85807 .03528 L .89935 .03361 L .93911 .03215 L .97619 .0309 L s 1 0 0 r .03353 .61803 m .04262 .5603 L .06244 .46143 L .08255 .38327 L .10458 .31721 L .12415 .27158 L .14509 .2331 L .16496 .20414 L .18653 .17904 L .22646 .1449 L .2463 .13215 L .26733 .12081 L .30517 .1046 L .34545 .0915 L .38422 .08166 L .42545 .07328 L .46516 .06668 L .50335 .06137 L .544 .05658 L .58313 .05262 L .62471 .04898 L .66477 .04592 L .70333 .04331 L .74433 .04087 L .78382 .03885 L .80368 .03801 L .82179 .03738 L .83181 .03711 L .83641 .03701 L .8413 .03692 L .84377 .03689 L .8465 .03686 L .84898 .03683 L .8501 .03683 L .85132 .03682 L .85265 .03681 L .85337 .03681 L .85405 .03681 L .85526 .03681 L .85656 .03681 L .85729 .03681 L .85808 .03681 L .85883 .03681 L .85952 .03682 L .86083 .03682 L .86221 .03683 L .86474 .03686 L .86714 .03689 L .87165 .03698 L .8765 .03712 L Mistroke .88179 .03732 L .88639 .03754 L .89129 .03784 L .9001 .03853 L .91041 .03966 L .92155 .04136 L .93207 .04352 L .94173 .0461 L .95817 .0521 L .9675 .05665 L .97619 .0618 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg@3oool00140oooo00@000000?ooo`3oool000002`3oool00`000000oooo0?ooo`0P0?ooo`040000 003oool0oooo000000X0oooo00<000000?ooo`3oool0803oool010000000oooo0?ooo`0000080?oo o`040000003oool0oooo00000280oooo00@000000?ooo`3oool00000203oool00`000000oooo0?oo o`0R0?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3oool0>03oool00140oooo00@0 00000?ooo`3oool000002`3oool00`000000oooo0?ooo`0P0?ooo`040000003oool0oooo000000L0 oooo1@00000Q0?ooo`040000003oool0oooo000000/0oooo00<000000?ooo`3oool0803oool01000 0000oooo0?ooo`0000090?ooo`030000003oool0oooo0240oooo00@000000?ooo`3oool000002P3o ool00`000000oooo0?ooo`0h0?ooo`002`3oool400000080oooo00@000000?ooo`3oool00000203o ool3000001d0oooo100000020?ooo`040000003oool0oooo000000L0oooo00@000000?ooo`3oool0 0000703oool400000080oooo00@000000?ooo`3oool000002@3oool2000001d0oooo100000020?oo o`040000003oool0oooo000000X0oooo00<000000?ooo`3oool06P3oool400000080oooo00@00000 0?ooo`3oool000002P3oool00`000000oooo0?ooo`0h0?ooo`004@3oool010000000oooo0?ooo`00 00080?ooo`030000003oool0oooo02<0oooo00@000000?ooo`3oool00000203oool00`000000oooo 0000000R0?ooo`040000003oool0oooo000000/0oooo00<000000?ooo`3oool0803oool010000000 oooo0?ooo`00000;0?ooo`030000003oool0oooo01l0oooo00@000000?ooo`3oool000002P3oool0 0`000000oooo0?ooo`0h0?ooo`004@3oool010000000oooo0?ooo`0000080?ooo`030000003oool0 oooo02<0oooo00@000000?ooo`3oool000002@3oool200000280oooo00@000000?ooo`3oool00000 203oool010000000oooo0?ooo`00000R0?ooo`040000003oool0oooo000000P0oooo00@000000?oo o`3oool000008@3oool010000000oooo0?ooo`0000080?ooo`<00000>P3oool00180oooo0P000009 0?ooo`@000008`3oool2000000/0oooo00<000000?ooo`3oool08@3oool2000000X0oooo0P00000T 0?ooo`8000002P3oool2000002<0oooo0P00000;0?ooo`030000003oool0oooo03P0oooo003o0?oo ob40oooo003o0?ooob40oooo000@0?ooool000001P00000;0?ooo`005P3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00H0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`060?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool01P3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo0100oooo003o0?ooo`h0oooo00<000000?ooo`3oool0403oool0 0?l0oooo3P3oool00`000000oooo0?ooo`0@0?ooo`00o`3oool>0?ooo`030000003oool0oooo0100 oooo003o0?ooo`h0oooo00<000000?ooo`3oool0403oool00?l0oooo3P3oool00`000000oooo0?oo o`0@0?ooo`00o`3oool>0?ooo`030000003oool0oooo0100oooo003o0?ooo`l0003o00<000000?oo o`3oool03`3oool00>00oooo0`000?lC0?l000T0003o3P3oool00`000000oooo0?ooo`0@0?ooo`00 b`3oool00`000?l0o`000?l0000E0?l001<0oooo203o000?0?ooo`030000003oool0oooo0100oooo 00310?ooo`030000o`3o0000o`0000P0o`000?ooo`80o`0000<0003o0?ooo`3oool0]@3oool00`000000oooo0?ooo`020?oo o`030000003oool0oooo0100oooo001<0?ooo`80o`000P000?ne0?ooo`040000003oool0oooo0000 00@0oooo00<000000?ooo`3oool0403oool004/0oooo00<0o`000000o`000?l0^03oool2000000D0 oooo00<000000?ooo`3oool0403oool004X0oooo00<0o`000000o`3oool0`03oool00`000000oooo 0?ooo`0@0?ooo`00B@3oool00`3o0000003o0?ooo`310?ooo`030000003oool0oooo0100oooo0018 0?ooo`030?l0003oool0oooo0<80oooo0P00000A0?ooo`00AP3oool20?l00P3oool00`3o0000003o0?ooo`3@0?ooo`030000003o ool0oooo0100oooo000j0?ooo`030?l0003oool0oooo0=00oooo00<000000?ooo`3oool0403oool0 03T0oooo00<0o`000000o`3oool0d@3oool200000140oooo000h0?ooo`030?l000000?l0oooo0=80 oooo00<000000?ooo`3oool0403oool003L0oooo00<0o`000?ooo`000?l0d`3oool00`000000oooo 0?ooo`0@0?ooo`00=`3oool00`3o0000003o0?ooo`3C0?ooo`030000003oool0oooo0100oooo000f 0?ooo`030?l0003oool0003o0=@0oooo00<000000?ooo`3oool0403oool003D0oooo00<0o`000?oo o`000?l0e@3oool00`000000oooo0?ooo`0@0?ooo`00=@3oool00`3o0000oooo0000o`3E0?ooo`03 0000003oool0oooo0100oooo000d0?ooo`030?l0003oool0003o0=00oooo00<000000?ooo`3oool0 0`3oool00`000000oooo0?ooo`0@0?ooo`00=03oool00`3o0000oooo0000o`3@0?ooo`030000003o ool0oooo00<0oooo0`00000@0?ooo`00<`3oool00`3o0000oooo0000o`3>0?ooo`D00000103oool0 0`000000oooo0?ooo`0@0?ooo`000?ooo`040000003oool0 oooo000000D0oooo00<000000?ooo`3oool0403oool00380oooo00<0o`000?ooo`000?l0d03oool0 0`000000oooo000000050?ooo`030000003oool0oooo0100oooo000a0?ooo`040?l0003oool0oooo 0000om40oooo0P0000050?ooo`030000003oool0oooo0100oooo000a0?ooo`030?l0003oool0003o 0=<0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`0@0?ooo`00<03oool0103o0000 oooo0?ooo`000?oI0?ooo`030000003oool0oooo0100oooo000`0?ooo`030?l0003oool0003o0=X0 oooo00<000000?ooo`3oool0403oool002l0oooo00@0o`000?ooo`3oool0003ofP3oool200000140 oooo000_0?ooo`030?l0003oool0003o0=/0oooo00<000000?ooo`3oool0403oool002h0oooo00@0 o`000?ooo`3oool0003of`3oool00`000000oooo0?ooo`0@0?ooo`00;P3oool00`3o0000oooo0000 o`3L0?ooo`030000003oool0oooo0100oooo000^0?ooo`030?l0003oool0003o0=`0oooo00<00000 0?ooo`3oool0403oool002d0oooo00@0o`000?ooo`3oool0003og03oool00`000000oooo0?ooo`0@ 0?ooo`00;@3oool0103o0000oooo0?ooo`000?oL0?ooo`030000003oool0oooo0100oooo000/0?oo o`040?l0003oool0oooo0000omd0oooo00<000000?ooo`3oool0403oool002`0oooo00@0o`000?oo o`3oool0003og@3oool200000140oooo000[0?ooo`050?l0003oool0oooo0?ooo`000?l0g@3oool0 0`000000oooo0?ooo`0@0?ooo`00:`3oool01@3o0000oooo0?ooo`3oool0003o0=d0oooo00<00000 0?ooo`3oool0403oool002/0oooo00@0o`000?ooo`3oool0003ogP3oool00`000000oooo0?ooo`0@ 0?ooo`00:P3oool01@3o0000oooo0?ooo`3oool0003o0=h0oooo00<000000?ooo`3oool0403oool0 02X0oooo00D0o`000?ooo`3oool0oooo0000o`3N0?ooo`030000003oool0oooo0100oooo000Z0?oo o`040?l0003oool0oooo0000oml0oooo00<000000?ooo`3oool0403oool002X0oooo00@0o`000?oo o`3oool0003og`3oool00`000000oooo0?ooo`0@0?ooo`00:@3oool01@3o0000oooo0?ooo`3oool0 003o0=l0oooo0P00000A0?ooo`00:@3oool01@3o0000oooo0?ooo`3oool0003o0=l0oooo00<00000 0?ooo`3oool0403oool002T0oooo00@0o`000?ooo`3oool0003oh03oool00`000000oooo0?ooo`0@ 0?ooo`00:03oool01@3o0000oooo0?ooo`3oool0003o0>00oooo00<000000?ooo`3oool0403oool0 02P0oooo00D0o`000?ooo`3oool0oooo0000o`3P0?ooo`030000003oool0oooo0100oooo000X0?oo o`040?l0003oool0oooo0000on40oooo00<000000?ooo`3oool0403oool002L0oooo00D0o`000?oo o`3oool0oooo0000o`3Q0?ooo`030000003oool0oooo0100oooo000W0?ooo`050?l0003oool0oooo 0?ooo`000?l0fP3oool2000000D0oooo00<000000?ooo`3oool0403oool002L0oooo00D0o`000?oo o`3oool0oooo0000o`3I0?ooo`040000003oool0oooo000000@0oooo0`00000@0?ooo`009`3oool0 103o0000oooo0?ooo`000?oJ0?ooo`040000003oool0oooo000000@0oooo00<000000?ooo`3oool0 403oool002H0oooo00D0o`000?ooo`3oool0oooo0000o`3J0?ooo`<000001@3oool00`000000oooo 0?ooo`0@0?ooo`009P3oool01@3o0000oooo0?ooo`3oool0003o0=X0oooo00<000000?ooo`3oool0 1@3oool00`000000oooo0?ooo`0@0?ooo`009P3oool01@3o0000oooo0?ooo`3oool0003o0=X0oooo 00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`0@0?ooo`009@3oool00`3o0000oooo0?oo o`020?ooo`030000o`3oool0oooo0=T0oooo0`0000040?ooo`030000003oool0oooo0100oooo000U 0?ooo`030?l0003oool0oooo0080oooo00<0003o0?ooo`3oool0h03oool00`000000oooo0?ooo`0@ 0?ooo`009@3oool01@3o0000oooo0?ooo`3oool0003o0><0oooo00<000000?ooo`3oool0403oool0 02D0oooo00D0o`000?ooo`3oool0oooo0000o`3S0?ooo`8000004@3oool002@0oooo00<0o`000?oo o`3oool00P3oool00`000?l0oooo0?ooo`3Q0?ooo`030000003oool0oooo0100oooo000T0?ooo`03 0?l0003oool0oooo0080oooo00<0003o0?ooo`3oool0h@3oool00`000000oooo0?ooo`0@0?ooo`00 903oool00`3o0000oooo0?ooo`020?ooo`030000o`3oool0oooo0>40oooo00<000000?ooo`3oool0 403oool002<0oooo00<0o`000?ooo`3oool00P3oool00`000?l0oooo0?ooo`3R0?ooo`030000003o ool0oooo0100oooo000S0?ooo`030?l0003oool0oooo0080oooo00<0003o0?ooo`3oool0hP3oool0 0`000000oooo0?ooo`0@0?ooo`008`3oool00`3o0000oooo0?ooo`020?ooo`030000o`3oool0oooo 0>80oooo00<000000?ooo`3oool0403oool00280oooo00<0o`000?ooo`3oool00`3oool00`000?l0 oooo0?ooo`3R0?ooo`030000003oool0oooo0100oooo000R0?ooo`030?l0003oool0oooo00<0oooo 00<0003o0?ooo`3oool0hP3oool200000140oooo000R0?ooo`030?l0003oool0oooo00<0oooo00<0 003o0?ooo`3oool0hP3oool00`000000oooo0?ooo`0@0?ooo`008P3oool00`3o0000oooo0?ooo`03 0?ooo`030000o`3oool0oooo0>80oooo00<000000?ooo`3oool0403oool00240oooo00<0o`000?oo o`3oool00`3oool00`000?l0oooo0?ooo`3S0?ooo`030000003oool0oooo0100oooo000Q0?ooo`03 0?l0003oool0oooo00<0oooo00<0003o0?ooo`3oool0h`3oool00`000000oooo0?ooo`0@0?ooo`00 8@3oool00`3o0000oooo0?ooo`030?ooo`030000o`3oool0oooo0><0oooo00<000000?ooo`3oool0 403oool00200oooo00<0o`000?ooo`3oool0103oool00`000?l0oooo0?ooo`3S0?ooo`030000003o ool0oooo0100oooo000P0?ooo`030?l0003oool0oooo00@0oooo00<0003o0?ooo`3oool0h`3oool0 0`000000oooo0?ooo`0@0?ooo`00803oool00`3o0000oooo0?ooo`040?ooo`030000o`3oool0oooo 0><0oooo0P00000A0?ooo`00803oool00`3o0000oooo0?ooo`040?ooo`030000o`3oool0oooo0><0 oooo00<000000?ooo`3oool0403oool001l0oooo00<0o`000?ooo`3oool01@3oool00`000?l0oooo 0?ooo`3S0?ooo`030000003oool0oooo0100oooo000O0?ooo`030?l0003oool0oooo00D0oooo00<0 003o0?ooo`3oool0h`3oool00`000000oooo0?ooo`0@0?ooo`007`3oool00`3o0000oooo0?ooo`05 0?ooo`030000o`3oool0oooo0><0oooo00<000000?ooo`3oool0403oool001l0oooo00<0o`000?oo o`3oool0103oool00`000?l0oooo0?ooo`3T0?ooo`030000003oool0oooo0100oooo000O0?ooo`03 0?l0003oool0oooo00@0oooo00<0003o0?ooo`3oool0i03oool00`000000oooo0?ooo`0@0?ooo`00 7P3oool00`3o0000oooo0?ooo`050?ooo`030000o`3oool0oooo0=d0oooo0P0000050?ooo`030000 003oool0oooo0100oooo000N0?ooo`030?l0003oool0oooo00D0oooo00<0003o0?ooo`3oool0g03o ool010000000oooo0?ooo`0000040?ooo`<00000403oool001h0oooo00<0o`000?ooo`3oool01@3o ool00`000?l0oooo0?ooo`3L0?ooo`040000003oool0oooo000000@0oooo00<000000?ooo`3oool0 403oool001h0oooo00<0o`000?ooo`3oool01@3oool00`000?l0oooo0?ooo`3M0?ooo`8000001@3o ool00`000000oooo0?ooo`0@0?ooo`007P3oool00`3o0000oooo0?ooo`050?ooo`030000o`3oool0 oooo0=`0oooo00@000000?ooo`3oool00000103oool00`000000oooo0?ooo`0@0?ooo`007P3oool0 0`3o0000oooo0?ooo`050?ooo`030000o`3oool0oooo0=`0oooo00@000000?ooo`3oool00000103o ool00`000000oooo0?ooo`0@0?ooo`007@3oool00`3o0000oooo0?ooo`060?ooo`030000o`3oool0 oooo0=d0oooo0P0000050?ooo`030000003oool0oooo0100oooo000M0?ooo`030?l0003oool0oooo 00D0oooo00<0003o0?ooo`3oool0i@3oool00`000000oooo0?ooo`0@0?ooo`007@3oool00`3o0000 oooo0?ooo`050?ooo`030000o`3oool0oooo0>D0oooo00<000000?ooo`3oool0403oool001d0oooo 00<0o`000?ooo`3oool01@3oool00`000?l0oooo0?ooo`3U0?ooo`8000004@3oool001d0oooo00<0 o`000?ooo`3oool01@3oool00`000?l0oooo0?ooo`3U0?ooo`030000003oool0oooo0100oooo000L 0?ooo`030?l0003oool0oooo00H0oooo00<0003o0?ooo`3oool0i@3oool00`000000oooo0?ooo`0@ 0?ooo`00703oool00`3o0000oooo0?ooo`060?ooo`030000o`3oool0oooo0>D0oooo00<000000?oo o`3oool0403oool001`0oooo00<0o`000?ooo`3oool01P3oool00`000?l0oooo0?ooo`3U0?ooo`03 0000003oool0oooo0100oooo000L0?ooo`030?l0003oool0oooo00H0oooo00<0003o0?ooo`3oool0 i@3oool00`000000oooo0?ooo`0@0?ooo`00703oool00`3o0000oooo0?ooo`060?ooo`030000o`3o ool0oooo0>D0oooo00<000000?ooo`3oool0403oool001/0oooo00<0o`000?ooo`3oool01P3oool0 0`000?l0oooo0?ooo`3V0?ooo`030000003oool0oooo0100oooo000K0?ooo`030?l0003oool0oooo 00H0oooo00<0003o0?ooo`3oool0iP3oool200000140oooo000K0?ooo`030?l0003oool0oooo00H0 oooo00<0003o0?ooo`3oool0iP3oool00`000000oooo0?ooo`0@0?ooo`006`3oool00`3o0000oooo 0?ooo`060?ooo`030000o`3oool0oooo0>H0oooo00<000000?ooo`3oool0403oool001/0oooo00<0 o`000?ooo`3oool01P3oool00`000?l0oooo0?ooo`3V0?ooo`030000003oool0oooo0100oooo000K 0?ooo`030?l0003oool0oooo00H0oooo00<0003o0?ooo`3oool0iP3oool00`000000oooo0?ooo`0@ 0?ooo`006P3oool00`3o0000oooo0?ooo`070?ooo`030000o`3oool0oooo0>H0oooo00<000000?oo o`3oool0403oool001X0oooo00<0o`000?ooo`3oool01`3oool00`000?l0oooo0?ooo`3V0?ooo`03 0000003oool0oooo0100oooo000J0?ooo`030?l0003oool0oooo00H0oooo00<0003o0?ooo`3oool0 i`3oool00`000000oooo0?ooo`0@0?ooo`006P3oool00`3o0000oooo0?ooo`060?ooo`030000o`3o ool0oooo0>L0oooo0P00000A0?ooo`006P3oool00`3o0000oooo0?ooo`060?ooo`030000o`3oool0 oooo0>L0oooo00<000000?ooo`3oool0403oool001X0oooo00<0o`000?ooo`3oool01P3oool00`00 0?l0oooo0?ooo`3W0?ooo`030000003oool0oooo0100oooo000J0?ooo`030?l0003oool0oooo00H0 oooo00<0003o0?ooo`3oool0i`3oool00`000000oooo0?ooo`0@0?ooo`006P3oool00`3o0000oooo 0?ooo`060?ooo`030000o`3oool0oooo0>L0oooo00<000000?ooo`3oool0403oool001T0oooo00<0 o`000?ooo`3oool01`3oool00`000?l0oooo0?ooo`3W0?ooo`030000003oool0oooo0100oooo000I 0?ooo`030?l0003oool0oooo00L0oooo00<0003o0?ooo`3oool0i`3oool00`000000oooo0?ooo`0@ 0?ooo`006@3oool00`3o0000oooo0?ooo`070?ooo`030000o`3oool0oooo0=T0oooo100000030?oo o`8000001@3oool00`000000oooo0?ooo`0@0?ooo`006@3oool00`3o0000oooo0?ooo`070?ooo`03 0000o`3oool0oooo0=/0oooo00D000000?ooo`3oool0oooo000000020?ooo`030000003oool0oooo 0080oooo0`00000@0?ooo`00o`3oool20?ooo`050000003oool0oooo0?ooo`0000000P3oool00`00 0000oooo0?ooo`0E0?ooo`00o`3oool20?ooo`050000003oool0oooo0?ooo`0000000P3oool00`00 0000oooo0?ooo`0E0?ooo`00o`3oool20?ooo`050000003oool0oooo0?ooo`0000000P3oool00`00 0000oooo0?ooo`0E0?ooo`00o`3oool3000000<0oooo00@000000?ooo`3oool000005`3oool00001 \ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.546552, -0.764114, 0.00202543, 0.0624232}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["9.", "Subsubsection"], Cell[TextData[{ "Find the radius and interval of convergence of the series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] x\^n\/\(n \@ n\)\)]], ".\n\nApply the Root Test: ", Cell[BoxData[ \(TraditionalForm \`\( | x\^n\/\(n \@ n\)\( | \^\(1/n\)\)\) = 1\/n\^\(3/\((2 n)\)\) | x | \[Rule] \( | x | \)\)]], " since ", Cell[BoxData[ \(TraditionalForm \`lim\_\(n \[Rule] \[Infinity]\)n\^\(3/\((2 n)\)\) = \(lim\_\(n \[Rule] \[Infinity]\)e\^\(3 \( ln(n)\)/\((2 n)\)\) = \(e\^0 = 1\)\)\)]], ". Therefore the series converges absolutely for ", Cell[BoxData[ \(TraditionalForm\`\( | x | \( < 1\)\)\)]], "and diverges for ", Cell[BoxData[ \(TraditionalForm\`\( | x | \( > 1\)\)\)]], ". The center is 0 and the radius of convergence is 1. At either endpoint ", Cell[BoxData[ \(TraditionalForm\`x = \(\[PlusMinus]1\)\)]], " the series converges absolutely since ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity] 1\/\(n \@ n\) = \[Sum]\+\(n = 1\)\%\[Infinity] 1\/n\^\(3/2\)\)]], " is a ", Cell[BoxData[ \(TraditionalForm\`p\)]], "-series with ", Cell[BoxData[ \(TraditionalForm\`p > 1\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p10\ = \ \[Sum]\+\(n = 1\)\%10 x\^n\/\(n \@ n\)\)], "Input"], Cell[BoxData[ \(x + x\^2\/\(2\ \@2\) + x\^3\/\(3\ \@3\) + x\^4\/8 + x\^5\/\(5\ \@5\) + x\^6\/\(6\ \@6\) + x\^7\/\(7\ \@7\) + x\^8\/\(16\ \@2\) + x\^9\/27 + x\^10\/\(10\ \@10\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[p10, {x, \(-1\)/2, 0}, PlotStyle \[Rule] RGBColor[1, 0, 0]]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.97619 1.90476 0.603319 1.36923 [ [.02381 .59082 -12 -9 ] [.02381 .59082 12 0 ] [.21429 .59082 -12 -9 ] [.21429 .59082 12 0 ] [.40476 .59082 -12 -9 ] [.40476 .59082 12 0 ] [.59524 .59082 -12 -9 ] [.59524 .59082 12 0 ] [.78571 .59082 -12 -9 ] [.78571 .59082 12 0 ] [.96369 .05563 -24 -4.5 ] [.96369 .05563 0 4.5 ] [.96369 .19255 -24 -4.5 ] [.96369 .19255 0 4.5 ] [.96369 .32947 -24 -4.5 ] [.96369 .32947 0 4.5 ] [.96369 .4664 -24 -4.5 ] [.96369 .4664 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .60332 m .02381 .60957 L s [(-0.5)] .02381 .59082 0 1 Mshowa .21429 .60332 m .21429 .60957 L s [(-0.4)] .21429 .59082 0 1 Mshowa .40476 .60332 m .40476 .60957 L s [(-0.3)] .40476 .59082 0 1 Mshowa .59524 .60332 m .59524 .60957 L s [(-0.2)] .59524 .59082 0 1 Mshowa .78571 .60332 m .78571 .60957 L s [(-0.1)] .78571 .59082 0 1 Mshowa .125 Mabswid .0619 .60332 m .0619 .60707 L s .1 .60332 m .1 .60707 L s .1381 .60332 m .1381 .60707 L s .17619 .60332 m .17619 .60707 L s .25238 .60332 m .25238 .60707 L s .29048 .60332 m .29048 .60707 L s .32857 .60332 m .32857 .60707 L s .36667 .60332 m .36667 .60707 L s .44286 .60332 m .44286 .60707 L s .48095 .60332 m .48095 .60707 L s .51905 .60332 m .51905 .60707 L s .55714 .60332 m .55714 .60707 L s .63333 .60332 m .63333 .60707 L s .67143 .60332 m .67143 .60707 L s .70952 .60332 m .70952 .60707 L s .74762 .60332 m .74762 .60707 L s .82381 .60332 m .82381 .60707 L s .8619 .60332 m .8619 .60707 L s .9 .60332 m .9 .60707 L s .9381 .60332 m .9381 .60707 L s .25 Mabswid 0 .60332 m 1 .60332 L s .97619 .05563 m .98244 .05563 L s [(-0.4)] .96369 .05563 1 0 Mshowa .97619 .19255 m .98244 .19255 L s [(-0.3)] .96369 .19255 1 0 Mshowa .97619 .32947 m .98244 .32947 L s [(-0.2)] .96369 .32947 1 0 Mshowa .97619 .4664 m .98244 .4664 L s [(-0.1)] .96369 .4664 1 0 Mshowa .125 Mabswid .97619 .08301 m .97994 .08301 L s .97619 .11039 m .97994 .11039 L s .97619 .13778 m .97994 .13778 L s .97619 .16516 m .97994 .16516 L s .97619 .21993 m .97994 .21993 L s .97619 .24732 m .97994 .24732 L s .97619 .2747 m .97994 .2747 L s .97619 .30209 m .97994 .30209 L s .97619 .35686 m .97994 .35686 L s .97619 .38424 m .97994 .38424 L s .97619 .41163 m .97994 .41163 L s .97619 .43901 m .97994 .43901 L s .97619 .49378 m .97994 .49378 L s .97619 .52116 m .97994 .52116 L s .97619 .54855 m .97994 .54855 L s .97619 .57593 m .97994 .57593 L s .97619 .02824 m .97994 .02824 L s .97619 .00086 m .97994 .00086 L s .25 Mabswid .97619 0 m .97619 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 1 0 0 r .5 Mabswid .02381 .01472 m .06244 .03557 L .10458 .05856 L .14415 .08039 L .18221 .10159 L .22272 .1244 L .26171 .1466 L .30316 .17046 L .34309 .19371 L .3815 .21633 L .42237 .24067 L .46172 .26438 L .49955 .28744 L .53984 .31229 L .57861 .3365 L .61984 .36256 L .65954 .38798 L .69774 .41275 L .73838 .43944 L .77751 .46549 L .81909 .49354 L .85916 .52096 L .89771 .54771 L .93871 .57656 L .97619 .60332 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg00oooo00@000000?ooo`3oool000001`3oool5000000D0oooo0`0000050?ooo`008`3oool20?l0 0=P0oooo100000020?ooo`040000003oool0oooo000000L0oooo00@000000?ooo`3oool000001P3o ool00`000000oooo0?ooo`050?ooo`009@3oool00`3o0000oooo0?ooo`3K0?ooo`040000003oool0 oooo000000P0oooo00<000000?ooo`0000001P3oool00`000000oooo0?ooo`050?ooo`009P3oool2 0?l00=/0oooo00@000000?ooo`3oool000002@3oool2000000H0oooo00<000000?ooo`3oool01@3o ool002P0oooo0P3o003J0?ooo`8000002`3oool00`000000oooo0?ooo`040?ooo`030000003oool0 oooo00D0oooo000Z0?ooo`030?l0003oool0oooo0>/0oooo00<000000?ooo`3oool01@3oool002/0 oooo0P3o003[0?ooo`030000003oool0oooo00D0oooo000]0?ooo`030?l0003oool0oooo0>P0oooo 0P0000060?ooo`00;P3oool20?l00>P0oooo00<000000?ooo`3oool01@3oool00300oooo0P3o003V 0?ooo`030000003oool0oooo00D0oooo000b0?ooo`80o`00i03oool00`000000oooo0?ooo`050?oo o`00=03oool20?l00>80oooo00<000000?ooo`3oool01@3oool003H0oooo0P3o003P0?ooo`030000 003oool0oooo00D0oooo000h0?ooo`030?l0003oool0oooo0=d0oooo00<000000?ooo`3oool01@3o ool003T0oooo0P3o003M0?ooo`030000003oool0oooo00D0oooo000k0?ooo`80o`00f`3oool20000 00H0oooo000m0?ooo`80o`00f@3oool00`000000oooo0?ooo`050?ooo`00?`3oool20?l00=L0oooo 00<000000?ooo`3oool01@3oool00440oooo0P3o003E0?ooo`030000003oool0oooo00D0oooo0013 0?ooo`80o`00d`3oool00`000000oooo0?ooo`050?ooo`00A@3oool20?l00=40oooo00<000000?oo o`3oool01@3oool004L0oooo0P3o003?0?ooo`030000003oool0oooo00D0oooo00190?ooo`80o`00 c@3oool00`000000oooo0?ooo`050?ooo`00B`3oool20?l000?ooo`03 0000003oool0oooo00D0oooo00280?ooo`80o`00SP3oool00`000000oooo0?ooo`050?ooo`00RP3o ool00`3o0000oooo0?ooo`2;0?ooo`030000003oool0oooo00D0oooo002;0?ooo`80o`00R`3oool2 000000H0oooo002=0?ooo`80o`00R@3oool00`000000oooo0?ooo`050?ooo`00S`3oool00`3o0000 oooo0?ooo`260?ooo`030000003oool0oooo00D0oooo002@0?ooo`80o`00QP3oool00`000000oooo 0?ooo`050?ooo`00TP3oool00`3o0000oooo0?ooo`230?ooo`030000003oool0oooo00D0oooo002C 0?ooo`80o`00P`3oool00`000000oooo0?ooo`050?ooo`00U@3oool00`3o0000oooo0?ooo`200?oo o`030000003oool0oooo00D0oooo002F0?ooo`80o`00P03oool00`000000oooo0?ooo`050?ooo`00 V03oool20?l007h0oooo0P0000060?ooo`00VP3oool00`3o0000oooo0?ooo`1k0?ooo`030000003o ool0oooo00D0oooo002K0?ooo`80o`00N`3oool00`000000oooo0?ooo`050?ooo`00W@3oool00`3o 0000oooo0?ooo`1h0?ooo`030000003oool0oooo00D0oooo002N0?ooo`80o`00N03oool00`000000 oooo0?ooo`050?ooo`00X03oool00`3o0000oooo0?ooo`1e0?ooo`030000003oool0oooo00D0oooo 002Q0?ooo`80o`00H@3oool2000000@0oooo00<000000?ooo`3oool00P3oool4000000D0oooo00<0 00000?ooo`3oool01@3oool00:<0oooo0P3o001N0?ooo`040000003oool0oooo000000P0oooo00<0 00000?ooo`3oool01P3oool00`000000oooo0?ooo`050?ooo`00Y@3oool00`3o0000oooo0?ooo`1K 0?ooo`040000003oool0oooo000000T0oooo00<000000?ooo`3oool01@3oool3000000D0oooo002V 0?ooo`80o`00E@3oool400000080oooo00@000000?ooo`3oool000002P3oool00`000000oooo0?oo o`040?ooo`030000003oool0oooo00D0oooo002X0?ooo`030?l0003oool0oooo05P0oooo00@00000 0?ooo`3oool000002`3oool00`000000oooo0?ooo`030?ooo`030000003oool0oooo00D0oooo002Y 0?ooo`80o`00F03oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo000000D0oooo 00<000000?ooo`3oool01@3oool00:/0oooo00<0o`000?ooo`3oool0EP3oool2000000X0oooo0P00 00060?ooo`030000003oool0oooo00D0oooo002/0?ooo`80o`00JP3oool00`000000oooo0?ooo`05 0?ooo`00[P3oool20?l006P0oooo00<000000?ooo`3oool01@3oool00;00oooo00<0o`000?ooo`3o ool0I@3oool00`000000oooo0?ooo`050?ooo`00/@3oool20?l006D0oooo0P0000060?ooo`00/`3o ool00`3o0000oooo0?ooo`1R0?ooo`030000003oool0oooo00D0oooo002d0?ooo`80o`00HP3oool0 0`000000oooo0?ooo`050?ooo`00]P3oool20?l00600oooo00<000000?ooo`3oool01@3oool00;P0 oooo00<0o`000?ooo`3oool0G@3oool00`000000oooo0?ooo`050?ooo`00^@3oool20?l005d0oooo 00<000000?ooo`3oool01@3oool00;/0oooo0P3o001K0?ooo`030000003oool0oooo00D0oooo002m 0?ooo`80o`00F@3oool2000000H0oooo002o0?ooo`030?l0003oool0oooo05H0oooo00<000000?oo o`3oool01@3oool00<00oooo0P3o001F0?ooo`030000003oool0oooo00D0oooo00320?ooo`030?l0 003oool0oooo05<0oooo00<000000?ooo`3oool01@3oool00<<0oooo00<0o`000?ooo`3oool0DP3o ool00`000000oooo0?ooo`050?ooo`00a03oool20?l00580oooo00<000000?ooo`3oool01@3oool0 0`3oool00`000000oooo0?ooo`05 0?ooo`00g@3oool00`3o0000oooo0?ooo`0h0?ooo`030000003oool0oooo00D0oooo003N0?ooo`80 o`00903oool2000000@0oooo00<000000?ooo`3oool00P3oool4000000D0oooo00<000000?ooo`3o ool01@3oool00>00oooo00<0o`000?ooo`3oool0803oool010000000oooo0?ooo`00000:0?ooo`03 0000003oool0oooo00@0oooo0`0000050?ooo`00h@3oool20?l00200oooo00@000000?ooo`3oool0 00002P3oool00`000000oooo0?ooo`040?ooo`030000003oool0oooo00D0oooo003S0?ooo`030?l0 003oool0oooo01L0oooo100000020?ooo`040000003oool0oooo000000X0oooo00<000000?ooo`3o ool0103oool00`000000oooo0?ooo`050?ooo`00i03oool20?l001d0oooo00@000000?ooo`3oool0 00002P3oool00`000000oooo0?ooo`040?ooo`030000003oool0oooo00D0oooo003V0?ooo`030?l0 003oool0oooo01X0oooo00@000000?ooo`3oool00000203oool3000000H0oooo00<000000?ooo`3o ool01@3oool00>L0oooo0P3o000K0?ooo`8000002`3oool00`000000oooo0?ooo`040?ooo`030000 003oool0oooo00D0oooo003Y0?ooo`030?l0003oool0oooo02`0oooo00<000000?ooo`3oool01@3o ool00>X0oooo0P3o000/0?ooo`030000003oool0oooo00D0oooo003/0?ooo`030?l0003oool0oooo 02T0oooo0P0000060?ooo`00k@3oool20?l002T0oooo00<000000?ooo`3oool01@3oool00>l0oooo 00<0o`000?ooo`3oool09P3oool00`000000oooo0?ooo`050?ooo`00l03oool00`3o0000oooo0?oo o`0U0?ooo`030000003oool0oooo00D0oooo003a0?ooo`80o`009@3oool00`000000oooo0?ooo`05 0?ooo`00l`3oool00`3o0000oooo0?ooo`0R0?ooo`030000003oool0oooo00D0oooo003d0?ooo`03 0?l0003oool0oooo0240oooo00<000000?ooo`3oool01@3oool00?D0oooo0P3o000Q0?ooo`030000 003oool0oooo00D0oooo003g0?ooo`030?l0003oool0oooo01h0oooo0P0000060?ooo`00n03oool2 0?l001h0oooo00<000000?ooo`3oool01@3oool00?X0oooo00<0o`000?ooo`3oool06`3oool00`00 0000oooo0?ooo`050?ooo`00n`3oool20?l001/0oooo00<000000?ooo`3oool01@3oool00?d0oooo 0P3o000I0?ooo`030000003oool0oooo00D0oooo003o0?ooo`030?l0003oool0oooo01H0oooo00<0 00000?ooo`3oool01@3oool00?l0oooo0@3oool20?l001H0oooo00<000000?ooo`3oool01@3oool0 0?l0oooo0`3oool00`3o0000oooo0?ooo`0C0?ooo`030000003oool0oooo00D0oooo003o0?ooo`@0 oooo0P3o000C0?ooo`8000001P3oool00?l0oooo1P3oool00`3o0000oooo0?ooo`0@0?ooo`030000 003oool0oooo00D0oooo003o0?ooo`L0oooo00<0o`000?ooo`3oool03`3oool00`000000oooo0?oo o`050?ooo`00o`3oool80?ooo`80o`003`3oool00`000000oooo0?ooo`050?ooo`00o`3oool:0?oo o`030?l0003oool0oooo00`0oooo00<000000?ooo`3oool01@3oool00?l0oooo2`3oool00`3o0000 oooo0?ooo`0;0?ooo`030000003oool0oooo00D0oooo00080?ooo`800000103oool00`000000oooo 0?ooo`020?ooo`<000009`3oool2000000@0oooo00<000000?ooo`3oool0103oool00`000000oooo 0?ooo`0V0?ooo`800000103oool00`000000oooo0?ooo`030?ooo`8000009`3oool2000000@0oooo 00<000000?ooo`3oool00P3oool4000002L0oooo0P0000040?ooo`030000003oool0oooo0080oooo 1000000N0?ooo`80o`002`3oool00`000000oooo0?ooo`050?ooo`001`3oool010000000oooo0?oo o`00000;0?ooo`030000003oool0oooo02<0oooo00@000000?ooo`3oool000002P3oool00`000000 oooo0?ooo`0U0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool000009@3oool0 10000000oooo0?ooo`0000080?ooo`030000003oool0oooo02L0oooo00@000000?ooo`3oool00000 2P3oool00`000000oooo0?ooo`0O0?ooo`030?l0003oool0oooo00P0oooo0P0000060?ooo`001`3o ool010000000oooo0?ooo`00000;0?ooo`030000003oool0oooo02<0oooo00@000000?ooo`3oool0 00001`3oool5000002H0oooo00@000000?ooo`3oool000002`3oool00`000000oooo0?ooo`0S0?oo o`040000003oool0oooo000000T0oooo00<000000?ooo`3oool09P3oool010000000oooo0?ooo`00 000:0?ooo`030000003oool0oooo0200oooo0P3o00080?ooo`030000003oool0oooo00D0oooo0000 0`3oool000000000000200000080oooo00@000000?ooo`3oool00000203oool300000200oooo1000 00020?ooo`040000003oool0oooo000000L0oooo00@000000?ooo`3oool000008@3oool400000080 oooo00@000000?ooo`3oool000002@3oool200000200oooo100000020?ooo`040000003oool0oooo 000000X0oooo00<000000?ooo`3oool07`3oool400000080oooo00@000000?ooo`3oool000002P3o ool00`000000oooo0?ooo`0R0?ooo`030?l0003oool0oooo00D0oooo00<000000?ooo`3oool01@3o ool000L0oooo00@000000?ooo`3oool00000203oool00`000000oooo0?ooo`0V0?ooo`040000003o ool0oooo000000P0oooo00<000000?ooo`0000009`3oool010000000oooo0?ooo`00000;0?ooo`03 0000003oool0oooo02<0oooo00@000000?ooo`3oool000002`3oool00`000000oooo0?ooo`0T0?oo o`040000003oool0oooo000000X0oooo00<000000?ooo`3oool08`3oool00`3o0000oooo0?ooo`04 0?ooo`030000003oool0oooo00D0oooo00070?ooo`040000003oool0oooo000000P0oooo00<00000 0?ooo`3oool09P3oool010000000oooo0?ooo`0000090?ooo`8000009`3oool010000000oooo0?oo o`0000080?ooo`040000003oool0oooo000002D0oooo00@000000?ooo`3oool00000203oool01000 0000oooo0?ooo`00000V0?ooo`040000003oool0oooo000000P0oooo0`00000V0?ooo`80o`00103o ool00`000000oooo0?ooo`050?ooo`00203oool2000000T0oooo1000000V0?ooo`8000002`3oool0 0`000000oooo0?ooo`0V0?ooo`8000002P3oool2000002L0oooo0P00000:0?ooo`800000:03oool2 000000/0oooo00<000000?ooo`3oool09P3oool01@3o0000oooo0?ooo`3oool0000000L0oooo003o 0?oooaH0oooo00@0o`000?ooo`3oool000001`3oool00?l0oooo5`3oool20?l000030000003oool0 oooo00D0oooo00050?ooool00000500000000`3o000000000000000500000000303oool00`000000 oooo0?ooo`070?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`070?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool01`3oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool01`3oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`070?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0203oool00`000000 oooo0?ooo`080?ooo`030000003oool0oooo00D0oooo000<0?ooo`030000003oool0oooo0380oooo 00<000000?ooo`3oool0<`3oool00`000000oooo0?ooo`0b0?ooo`030000003oool0oooo03<0oooo 00<000000?ooo`3oool0<`3oool00`000000oooo0?ooo`050?ooo`00o`3ooolI0?ooo`030000003o ool0oooo00D0oooo003o0?oooaT0oooo00<000000?ooo`3oool01@3oool00?l0oooo8@3oool00001 \ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.522366, -0.444302, 0.00186366, 0.00259256}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["13.", "Subsubsection"], Cell[TextData[{ "Find the radius and interval of convergence of the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity] x\^\(2 n + 1\)\/\(n!\)\)]], ".\n\nApply the Ratio Test: ", Cell[BoxData[ \(TraditionalForm \`\( | a\_\(n + 1\)/a\_n | \) = \(\( | \(x\^\(2 n + 3\)\/\(\((n + 1)\)!\)\)/ \(x\^\(2 n + 1\)\/\(n!\)\) | \) = \( | x\( | \^2\)\)\/\((n + 1)\) \[Rule] 0\)\)]], " for any ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". Therefore the series converges absolutely for all ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". The center is 0 and the radius of convergence is ", Cell[BoxData[ \(TraditionalForm\`\[Infinity]\)]], ". We shall soon see that ", Cell[BoxData[ \(TraditionalForm \`e\^u = \ \[Sum]\+\(n = 0\)\%\[Infinity] u\^n\/\(n!\)\)]], "for all ", Cell[BoxData[ \(TraditionalForm\`u\)]], " (see also Problem 42) and the given series can be written in terms of \ this: ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity] x\^\(2 n + 1\)\/\(n!\) = \(x \(\[Sum]\+\(n = 0\)\%\[Infinity]\((x\^2)\)\^n\/\(n!\)\) = x\ e\^\(x\^2\)\)\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p10\ = \ \[Sum]\+\(n = 1\)\%10 x\^\(2 n + 1\)\/\(n!\)\)], "Input"], Cell[BoxData[ \(x\^3 + x\^5\/2 + x\^7\/6 + x\^9\/24 + x\^11\/120 + x\^13\/720 + x\^15\/5040 + x\^17\/40320 + x\^19\/362880 + x\^21\/3628800\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{x\ E\^\(x\^2\), p10}, \ {x, \(-2\), 2}, PlotStyle \[Rule] {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.238095 0.315636 0.0182232 [ [.02381 .30314 -6 -9 ] [.02381 .30314 6 0 ] [.2619 .30314 -6 -9 ] [.2619 .30314 6 0 ] [.7381 .30314 -3 -9 ] [.7381 .30314 3 0 ] [.97619 .30314 -3 -9 ] [.97619 .30314 3 0 ] [.4875 .04229 -18 -4.5 ] [.4875 .04229 0 4.5 ] [.4875 .1334 -18 -4.5 ] [.4875 .1334 0 4.5 ] [.4875 .22452 -12 -4.5 ] [.4875 .22452 0 4.5 ] [.4875 .40675 -6 -4.5 ] [.4875 .40675 0 4.5 ] [.4875 .49787 -12 -4.5 ] [.4875 .49787 0 4.5 ] [.4875 .58898 -12 -4.5 ] [.4875 .58898 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 .31564 m .02381 .32189 L s [(-2)] .02381 .30314 0 1 Mshowa .2619 .31564 m .2619 .32189 L s [(-1)] .2619 .30314 0 1 Mshowa .7381 .31564 m .7381 .32189 L s [(1)] .7381 .30314 0 1 Mshowa .97619 .31564 m .97619 .32189 L s [(2)] .97619 .30314 0 1 Mshowa .125 Mabswid .07143 .31564 m .07143 .31939 L s .11905 .31564 m .11905 .31939 L s .16667 .31564 m .16667 .31939 L s .21429 .31564 m .21429 .31939 L s .30952 .31564 m .30952 .31939 L s .35714 .31564 m .35714 .31939 L s .40476 .31564 m .40476 .31939 L s .45238 .31564 m .45238 .31939 L s .54762 .31564 m .54762 .31939 L s .59524 .31564 m .59524 .31939 L s .64286 .31564 m .64286 .31939 L s .69048 .31564 m .69048 .31939 L s .78571 .31564 m .78571 .31939 L s .83333 .31564 m .83333 .31939 L s .88095 .31564 m .88095 .31939 L s .92857 .31564 m .92857 .31939 L s .25 Mabswid 0 .31564 m 1 .31564 L s .5 .04229 m .50625 .04229 L s [(-15)] .4875 .04229 1 0 Mshowa .5 .1334 m .50625 .1334 L s [(-10)] .4875 .1334 1 0 Mshowa .5 .22452 m .50625 .22452 L s [(-5)] .4875 .22452 1 0 Mshowa .5 .40675 m .50625 .40675 L s [(5)] .4875 .40675 1 0 Mshowa .5 .49787 m .50625 .49787 L s [(10)] .4875 .49787 1 0 Mshowa .5 .58898 m .50625 .58898 L s [(15)] .4875 .58898 1 0 Mshowa .125 Mabswid .5 .06051 m .50375 .06051 L s .5 .07874 m .50375 .07874 L s .5 .09696 m .50375 .09696 L s .5 .11518 m .50375 .11518 L s .5 .15163 m .50375 .15163 L s .5 .16985 m .50375 .16985 L s .5 .18807 m .50375 .18807 L s .5 .2063 m .50375 .2063 L s .5 .24274 m .50375 .24274 L s .5 .26097 m .50375 .26097 L s .5 .27919 m .50375 .27919 L s .5 .29741 m .50375 .29741 L s .5 .33386 m .50375 .33386 L s .5 .35208 m .50375 .35208 L s .5 .37031 m .50375 .37031 L s .5 .38853 m .50375 .38853 L s .5 .42498 m .50375 .42498 L s .5 .4432 m .50375 .4432 L s .5 .46142 m .50375 .46142 L s .5 .47964 m .50375 .47964 L s .5 .51609 m .50375 .51609 L s .5 .53431 m .50375 .53431 L s .5 .55254 m .50375 .55254 L s .5 .57076 m .50375 .57076 L s .5 .02407 m .50375 .02407 L s .5 .00584 m .50375 .00584 L s .5 .60721 m .50375 .60721 L s .25 Mabswid .5 0 m .5 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .5 Mabswid .13086 0 m .14603 .06861 L .16586 .13235 L .18689 .18055 L .20643 .21287 L .22473 .23544 L .24497 .25416 L .2635 .26708 L .28248 .27728 L .30321 .28581 L .32288 .29206 L .34385 .29726 L .36383 .30118 L .38544 .30458 L .4255 .30935 L .4665 .31302 L .50599 .3161 L .54396 .31912 L .58439 .32296 L .6233 .32798 L .64454 .33163 L .66466 .33597 L .68457 .3414 L .70299 .34777 L .72165 .35599 L .74225 .36785 L .7616 .38259 L .78246 .40395 L .8025 .43194 L .8236 .47271 L .84334 .52586 L .86171 .59394 L s .86171 .59394 m .86634 .61803 L s 1 0 0 r .12484 0 m .12589 .00613 L .14603 .0957 L .16586 .15793 L .18689 .20451 L .20643 .23534 L .22473 .25651 L .24497 .27368 L .2635 .28519 L .28248 .29393 L .30321 .30087 L .32288 .30562 L .34385 .30921 L .35414 .31055 L .36526 .31174 L .37488 .31259 L .38544 .31335 L .39506 .31391 L .40537 .3144 L .41627 .31479 L .42644 .31507 L .43661 .31528 L .44236 .31537 L .44768 .31544 L .45743 .31553 L .46236 .31556 L .4678 .31559 L .47272 .31561 L .47519 .31562 L .47789 .31562 L .48021 .31563 L .48269 .31563 L .48502 .31563 L .48715 .31563 L .48845 .31563 L .48982 .31564 L .491 .31564 L .49229 .31564 L .49368 .31564 L .49446 .31564 L .49518 .31564 L .49584 .31564 L .49656 .31564 L .49787 .31564 L .49856 .31564 L .49921 .31564 L .50042 .31564 L .50172 .31564 L .50246 .31564 L .50313 .31564 L Mistroke .50426 .31564 L .5055 .31564 L .5068 .31564 L .508 .31564 L .50927 .31564 L .51061 .31564 L .51302 .31564 L .51436 .31564 L .51562 .31564 L .51848 .31565 L .52097 .31565 L .52326 .31565 L .52839 .31567 L .53272 .31568 L .53745 .31571 L .54238 .31574 L .54696 .31578 L .55203 .31583 L .55751 .3159 L .56746 .31607 L .57714 .31629 L .58606 .31656 L .59578 .31692 L .60486 .31735 L .61546 .31798 L .62552 .31871 L .63525 .31958 L .64584 .32072 L .66439 .32332 L .68414 .32718 L .70514 .33292 L .72459 .3403 L .74286 .34966 L .76302 .36371 L .78152 .38129 L .80042 .40562 L .82111 .44257 L .84069 .49161 L .86164 .56595 L .87186 .61346 L Mfstroke .87186 .61346 m .87268 .61803 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg0?ooo`009@3oool00`3o0000003o0?ooo`1W 0?ooo`030000003oool0oooo08h0oooo000U0?ooo`030?l0003oool0003o06L0oooo0P00002?0?oo o`009@3oool00`3o0000oooo0000o`1W0?ooo`030000003oool0oooo08h0oooo000V0?ooo`030?l0 00000?l0oooo06H0oooo00<000000?ooo`3oool0SP3oool002H0oooo00<0o`000000o`3oool0IP3o ool00`000000oooo0?ooo`2>0?ooo`009P3oool00`3o0000oooo0000o`1G0?ooo`@000000P3oool3 000000H0oooo00<000000?ooo`3oool0SP3oool002H0oooo00<0o`000?ooo`000?l0F@3oool00`00 0000oooo0?ooo`040?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0SP3oool002L0 oooo00<0o`000000o`3oool0F03oool00`000000oooo0?ooo`040?ooo`030000003oool0oooo00<0 oooo0`00002>0?ooo`009`3oool00`3o0000003o0?ooo`1@0?ooo`@00000103oool010000000oooo 0?ooo`3oool3000000H0oooo00<000000?ooo`3oool0SP3oool002L0oooo00<0o`000?ooo`000?l0 F03oool01@000000oooo0?ooo`3oool0000000P0oooo00<000000?ooo`3oool0SP3oool002L0oooo 00<0o`000?ooo`000?l0EP3oool3000000<0oooo00<000000?ooo`3oool01P3oool00`000000oooo 0?ooo`2>0?ooo`009`3oool00`3o0000oooo0000o`1H0?ooo`040000003oool0oooo0?ooo`@00000 1@3oool00`000000oooo0?ooo`2>0?ooo`00:03oool00`3o0000003o0?ooo`1T0?ooo`800000S`3o ool002P0oooo00<0o`000?ooo`000?l0I03oool00`000000oooo0?ooo`2>0?ooo`00:03oool00`3o 0000oooo0000o`1T0?ooo`030000003oool0oooo08h0oooo000X0?ooo`030?l0003oool0003o06@0 oooo00<000000?ooo`3oool0SP3oool002T0oooo00<0o`000?ooo`000?l0H`3oool00`000000oooo 0?ooo`2>0?ooo`00:@3oool00`3o0000oooo0000o`1S0?ooo`800000S`3oool002T0oooo00<0o`00 0?ooo`000?l0H`3oool00`000000oooo0?ooo`2>0?ooo`00:@3oool00`3o0000oooo0000o`1S0?oo o`030000003oool0oooo08h0oooo000Z0?ooo`030?l0003oool0003o0680oooo00<000000?ooo`3o ool0SP3oool002X0oooo00<0o`000?ooo`000?l0HP3oool00`000000oooo0?ooo`2>0?ooo`00:P3o ool00`3o0000oooo0000o`1R0?ooo`800000S`3oool002X0oooo00<0o`000?ooo`000?l0HP3oool0 0`000000oooo0?ooo`2>0?ooo`00:`3oool00`3o0000oooo0000o`1Q0?ooo`030000003oool0oooo 08h0oooo000[0?ooo`030?l0003oool0003o0640oooo00<000000?ooo`3oool0SP3oool002/0oooo 00<0o`000?ooo`000?l0H@3oool00`000000oooo0?ooo`2>0?ooo`00:`3oool0103o0000oooo0?oo o`000?mP0?ooo`030000003oool0oooo08h0oooo000/0?ooo`030?l0003oool0003o0600oooo0P00 002?0?ooo`00;03oool00`3o0000oooo0000o`1P0?ooo`030000003oool0oooo08h0oooo000/0?oo o`030?l0003oool0003o0600oooo00<000000?ooo`3oool0SP3oool002`0oooo00@0o`000?ooo`3o ool0003oD03oool4000000<0oooo0P0000060?ooo`030000003oool0oooo08h0oooo000]0?ooo`03 0?l0003oool0003o0580oooo00D000000?ooo`3oool0oooo000000020?ooo`030000003oool0oooo 00<0oooo00<000000?ooo`3oool0SP3oool002d0oooo00<0o`000?ooo`000?l0DP3oool01@000000 oooo0?ooo`3oool000000080oooo00<000000?ooo`3oool00`3oool3000008h0oooo000]0?ooo`04 0?l0003oool0oooo0000odT0oooo100000040?ooo`050000003oool0oooo0?ooo`0000000P3oool0 0`000000oooo0?ooo`030?ooo`030000003oool0oooo08h0oooo000^0?ooo`030?l0003oool0003o 0540oooo00D000000?ooo`3oool0oooo000000020?ooo`030000003oool0oooo00<0oooo00<00000 0?ooo`3oool0SP3oool002h0oooo00@0o`000?ooo`3oool0003oCP3oool3000000<0oooo00@00000 0?ooo`3oool000001@3oool00`000000oooo0?ooo`2>0?ooo`00;P3oool0103o0000oooo0?ooo`00 0?m@0?ooo`030000003oool0oooo0080oooo0P0000060?ooo`030000003oool0oooo08h0oooo000^ 0?ooo`050?l0003oool0oooo0?ooo`000?l0G03oool2000008l0oooo000_0?ooo`040?l0003oool0 oooo0000oe`0oooo00<000000?ooo`3oool0SP3oool002l0oooo00@0o`000?ooo`3oool0003oG03o ool00`000000oooo0?ooo`2>0?ooo`00;`3oool01@3o0000oooo0?ooo`3oool0003o05/0oooo00<0 00000?ooo`3oool0SP3oool00300oooo00@0o`000?ooo`3oool0003oF`3oool00`000000oooo0?oo o`2>0?ooo`00<03oool01@3o0000oooo0?ooo`3oool0003o05X0oooo0P00002?0?ooo`00<@3oool0 103o0000oooo0?ooo`000?mJ0?ooo`030000003oool0oooo08h0oooo000a0?ooo`050?l0003oool0 oooo0?ooo`000?l0F@3oool00`000000oooo0?ooo`2>0?ooo`000?ooo`00<`3oool0103o0000oooo0?ooo`000?mH0?ooo`800000S`3o ool003<0oooo00D0o`000?ooo`3oool0oooo0000o`1G0?ooo`030000003oool0oooo08h0oooo000d 0?ooo`040?l0003oool0oooo0000oeL0oooo00<000000?ooo`3oool0SP3oool003@0oooo00D0o`00 0?ooo`3oool0oooo0000o`1F0?ooo`030000003oool0oooo08h0oooo000e0?ooo`050?l0003oool0 oooo0?ooo`000?l0E@3oool00`000000oooo0?ooo`2>0?ooo`00=@3oool01@3o0000oooo0?ooo`3o ool0003o05D0oooo00<000000?ooo`3oool0SP3oool003H0oooo00D0o`000?ooo`3oool0oooo0000 o`1D0?ooo`800000S`3oool003H0oooo00D0o`000?ooo`3oool0oooo0000o`1D0?ooo`030000003o ool0oooo08h0oooo000g0?ooo`050?l0003oool0oooo0?ooo`000?l0D`3oool00`000000oooo0?oo o`2>0?ooo`00>03oool01@3o0000oooo0?ooo`3oool0003o0580oooo00<000000?ooo`3oool0SP3o ool003P0oooo00<0o`000?ooo`3oool00P3oool00`000?l0oooo0?ooo`160?ooo`<000001P3oool0 0`000000oooo0?ooo`2>0?ooo`00>@3oool01@3o0000oooo0?ooo`3oool0003o04/0oooo00<00000 0?ooo`3oool00`3oool3000008h0oooo000j0?ooo`050?l0003oool0oooo0?ooo`000?l0BP3oool0 0`000000oooo0?ooo`030?ooo`030000003oool0oooo08h0oooo000j0?ooo`030?l0003oool0oooo 0080oooo00<0003o0?ooo`3oool0?P3oool400000080oooo0`0000060?ooo`030000003oool0oooo 08h0oooo000k0?ooo`030?l0003oool0oooo0080oooo00<0003o0?ooo`3oool0@`3oool00`000000 oooo0?ooo`060?ooo`030000003oool0oooo08h0oooo000l0?ooo`030?l0003oool0oooo0080oooo 00<0003o0?ooo`3oool0@P3oool00`000000oooo0?ooo`060?ooo`030000003oool0oooo08h0oooo 000m0?ooo`030?l0003oool0oooo0080oooo00<0003o0?ooo`3oool0@@3oool4000000D0oooo0P00 002?0?ooo`00?@3oool00`3o0000oooo0?ooo`030?ooo`80003oBP3oool00`000000oooo0?ooo`2> 0?ooo`00?P3oool00`3o0000oooo0?ooo`040?ooo`030000o`3oool0oooo04L0oooo00<000000?oo o`3oool0SP3oool003l0oooo00<0o`000?ooo`3oool0103oool00`000?l0oooo0?ooo`160?ooo`03 0000003oool0oooo08h0oooo00100?ooo`030?l0003oool0oooo00@0oooo00<0003o0?ooo`3oool0 A@3oool00`000000oooo0?ooo`2>0?ooo`00@@3oool00`3o0000oooo0?ooo`040?ooo`80003oA@3o ool2000008l0oooo00120?ooo`030?l0003oool0oooo00D0oooo00<0003o0?ooo`3oool0@P3oool0 0`000000oooo0?ooo`2>0?ooo`00@`3oool20?l000H0oooo00<0003o0?ooo`3oool0@@3oool00`00 0000oooo0?ooo`2>0?ooo`00A@3oool00`3o0000oooo0?ooo`040?ooo`80003o@@3oool00`000000 oooo0?ooo`2>0?ooo`00AP3oool00`3o0000oooo0?ooo`050?ooo`80003o?`3oool00`000000oooo 0?ooo`2>0?ooo`00A`3oool20?l000L0oooo0`000?ll0?ooo`800000S`3oool004T0oooo0P3o0008 0?ooo`<0003o>@3oool00`000000oooo0?ooo`2>0?ooo`00203oool4000003l0oooo00<0o`000000 000000000P0000060?ooo`<0003o=P3oool00`000000oooo0?ooo`0o0?ooo`@00000@03oool40000 00L0oooo00080?ooo`030000003oool0oooo0440oooo0P3o00000`000000oooo0?ooo`080?ooo`80 003o=03oool00`000000oooo0?ooo`110?ooo`030000003oool0oooo03l0oooo00<000000?ooo`3o ool0203oool000T0oooo00<000000?ooo`3oool0@P3oool20?l000/0oooo0`000?la0?ooo`030000 003oool0oooo0440oooo00<000000?ooo`3oool0@03oool00`000000oooo0?ooo`070?ooo`000P3o ool4000000@0oooo00<000000?ooo`3oool0>@3oool4000000@0oooo00<000000?ooo`3o00000P3o 000;0?ooo`<0003o;P3oool00`000000oooo0?ooo`110?ooo`030000003oool0oooo0440oooo00<0 00000?ooo`3oool01P3oool000/0oooo00<000000?ooo`3oool0@03oool00`000000oooo0?ooo`02 0?ooo`<0o`002`3oool40000obX0oooo0P0000120?ooo`030000003oool0oooo0480oooo00<00000 0?ooo`3oool01@3oool000P0oooo00@000000?ooo`3oool00000@03oool3000000L0oooo103o000; 0?ooo`H0003o903oool00`000000oooo0?ooo`0o0?ooo`<00000@@3oool010000000oooo0?ooo`00 00070?ooo`002@3oool2000004<0oooo00<000000?ooo`3oool02@3oool50?l000`0oooo2@000?lK 0?ooo`030000003oool0oooo0440oooo00<000000?ooo`3oool0@03oool2000000P0oooo001O0?oo o`P0o`003@3oool<0000o`l0oooo00<000000?ooo`3oool0SP3oool006L0oooo303o000=0?ooo`/0 003o103oool00`000000oooo0?ooo`2>0?ooo`00L`00000l0?l0070000000@3oool000L0oooo00<0 00000?ooo`3oool02P3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo00/0oooo00<0 00000?ooo`3oool02P3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo00X0oooo00<0 00000?ooo`3oool02`3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo00X0oooo00<0 00000?ooo`3oool02`3oool00`000000oooo0?ooo`050?ooo`/0003o203oool00`000000oooo0?oo o`020?ooo`d0o`002P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00/0oooo00<0 00000?ooo`3oool02P3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo00/0oooo00<0 00000?ooo`3oool02P3oool00`000000oooo0?ooo`060?ooo`001`3oool00`000000oooo0?ooo`11 0?ooo`030000003oool0oooo0440oooo00<000000?ooo`3oool0403oool;0000o`l0oooo1@3o000B 0?ooo`030000003oool0oooo0440oooo00<000000?ooo`3oool01P3oool008l0oooo00<000000?oo o`3oool06`3oool70000o`d0oooo1@3o001J0?ooo`00S`3oool00`000000oooo0?ooo`0R0?ooo`<0 003o3`3oool30?l005L0oooo002?0?ooo`8000009P3oool40000o`h0oooo0`3o001D0?ooo`00S`3o ool00`000000oooo0?ooo`0Y0?ooo`H0003o2`3oool20?l00580oooo002?0?ooo`030000003oool0 oooo02l0oooo1@000?l80?ooo`80o`00D03oool008l0oooo00<000000?ooo`3oool0=03oool20000 o`P0oooo0P3o001>0?ooo`00S`3oool00`000000oooo0?ooo`0f0?ooo`<0003o1`3oool20?l004`0 oooo002?0?ooo`800000>P3oool20000o`L0oooo0P3o001:0?ooo`00S`3oool00`000000oooo0?oo o`0k0?ooo`80003o1`3oool00`3o0000oooo0?ooo`170?ooo`00S`3oool00`000000oooo0?ooo`0m 0?ooo`80003o1P3oool20?l004L0oooo002?0?ooo`030000003oool0oooo03l0oooo00<0003o0?oo o`3oool01@3oool00`3o0000oooo0?ooo`140?ooo`00S`3oool00`000000oooo0?ooo`100?ooo`80 003o1P3oool00`3o0000oooo0?ooo`130?ooo`00S`3oool00`000000oooo0?ooo`120?ooo`030000 o`3oool0oooo00@0oooo00<0o`000?ooo`3oool0@P3oool008l0oooo0P0000140?ooo`80003o1@3o ool00`3o0000oooo0?ooo`110?ooo`00S`3oool00`000000oooo0?ooo`150?ooo`030000o`3oool0 oooo00<0oooo0P3o00110?ooo`00S`3oool00`000000oooo0?ooo`160?ooo`80003o1@3oool00`3o 0000oooo0?ooo`0n0?ooo`00S`3oool00`000000oooo0?ooo`180?ooo`030000o`3oool0oooo00<0 oooo00<0o`000?ooo`3oool0?@3oool008l0oooo00<000000?ooo`3oool0B@3oool00`000?l0oooo 0?ooo`030?ooo`030?l0003oool0oooo03`0oooo002?0?ooo`800000B`3oool00`000?l0oooo0?oo o`020?ooo`030?l0003oool0oooo03`0oooo002?0?ooo`030000003oool0oooo04/0oooo00<0003o 0?ooo`3oool00P3oool00`3o0000oooo0?ooo`0k0?ooo`00S`3oool00`000000oooo0?ooo`1<0?oo o`030000o`3oool0oooo0080oooo00<0o`000?ooo`3oool0>P3oool008l0oooo00<000000?ooo`3o ool0C@3oool00`000?l0oooo0?ooo`020?ooo`030?l0003oool0oooo03T0oooo00260?ooo`<00000 1P3oool00`000000oooo0?ooo`1>0?ooo`050000o`3oool0oooo0?ooo`3o0000>`3oool008T0oooo 00<000000?ooo`3oool00`3oool3000004l0oooo00D0003o0?ooo`3oool0oooo0?l0000j0?ooo`00 R@3oool00`000000oooo0?ooo`030?ooo`030000003oool0oooo04l0oooo00<0003o0?ooo`3oool0 0P3oool00`3o0000oooo0?ooo`0g0?ooo`00QP3oool3000000H0oooo00<000000?ooo`3oool0D03o ool01@000?l0oooo0?ooo`3oool0o`0003T0oooo00260?ooo`030000003oool0oooo00H0oooo00<0 00000?ooo`3oool0D@3oool01@000?l0oooo0?ooo`3oool0o`0003P0oooo00260?ooo`030000003o ool0oooo00H0oooo00<000000?ooo`3oool0DP3oool010000?l0oooo0?ooo`3o000h0?ooo`00QP3o ool4000000D0oooo0P00001C0?ooo`050000o`3oool0oooo0?ooo`3o0000=`3oool008l0oooo00<0 00000?ooo`3oool0D`3oool01@000?l0oooo0?ooo`3oool0o`0003H0oooo002?0?ooo`030000003o ool0oooo05@0oooo00@0003o0?ooo`3oool0o`00=P3oool008l0oooo00<000000?ooo`3oool0E03o ool01@000?l0oooo0?ooo`3oool0o`0003D0oooo002?0?ooo`030000003oool0oooo05D0oooo00@0 003o0?ooo`3oool0o`00=@3oool008l0oooo0P00001F0?ooo`050000o`3oool0oooo0?ooo`3o0000 =03oool008l0oooo00<000000?ooo`3oool0EP3oool010000?l0oooo0?ooo`3o000d0?ooo`00S`3o ool00`000000oooo0?ooo`1F0?ooo`050000o`3oool0oooo0?ooo`3o0000<`3oool008l0oooo00<0 00000?ooo`3oool0E`3oool010000?l0oooo0?ooo`3o000c0?ooo`00S`3oool00`000000oooo0?oo o`1G0?ooo`050000o`3oool0oooo0?ooo`3o0000"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-2.10521, -17.3207, 0.0146705, 0.191677}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["33.", "Subsubsection"], Cell[TextData[{ "Find the radius and interval of convergence of the series ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity]\((x - 1)\)\^\(2 n\)\/4\^n\)]], " and find the sum of the series in this interval.\n\nThe series is \ geometric: ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity]\((x - 1)\)\^\(2 n\)\/4\^n\)]], "= ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 0\)\%\[Infinity] r\^n\)]], " where ", Cell[BoxData[ \(TraditionalForm\`r = \((x - 1)\)\^2\/4\)]], ". Therefore it converges absolutely for ", Cell[BoxData[ \(TraditionalForm\`\( | \((x - 1)\)\^2\/4 | \( < 1\)\)\)]], " and diverges for ", Cell[BoxData[ \(TraditionalForm \`\( | \((x - 1)\)\^2\/4 | \( \[GreaterEqual] 1\)\)\)]], ". The first inequality is equivalent to ", Cell[BoxData[ \(TraditionalForm\`\( | x - 1\( | \^2\)\( < 4\)\)\)]], " or ", Cell[BoxData[ \(TraditionalForm\`\( | x - 1 | \( < 2\)\)\)]], " which shows that the center is 1 and the radius is 2. The interval of \ convergence is thus ", Cell[BoxData[ \(TraditionalForm\`\(-1\) < x < 3\)]], ". Usingthe fromula for a geometric series we find that ", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 0\)\%\[Infinity]\((x - 1)\)\^\(2 n\)\/4\^n = \(1\/\(1 - \((x - 1)\)\^2\/4\) = \(4\/\(4 - \((x - 1)\)\^2\) = 4\/\(3 + 2 x - x\^2\)\)\)\)]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(p10\ = \ \[Sum]\+\(n = 0\)\%10\((x - 1)\)\^\(2 n\)\/4\^n\)], "Input"], Cell[BoxData[ \(1 + 1\/4\ \((\(-1\) + x)\)\^2 + 1\/16\ \((\(-1\) + x)\)\^4 + 1\/64\ \((\(-1\) + x)\)\^6 + 1\/256\ \((\(-1\) + x)\)\^8 + \((\(-1\) + x)\)\^10\/1024 + \((\(-1\) + x)\)\^12\/4096 + \((\(-1\) + x)\)\^14\/16384 + \((\(-1\) + x)\)\^16\/65536 + \((\(-1\) + x)\)\^18\/262144 + \((\(-1\) + x)\)\^20\/1048576\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Plot[{4\/\(3 + 2 x - x\^2\), p10}, \ {x, \(-1\), 3}, PlotRange \[Rule] {0, 10}, PlotStyle \[Rule] {RGBColor[0, 0, 1], RGBColor[1, 0, 0]}]\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.261905 0.238095 0 0.0618034 [ [.02381 -0.0125 -6 -9 ] [.02381 -0.0125 6 0 ] [.5 -0.0125 -3 -9 ] [.5 -0.0125 3 0 ] [.7381 -0.0125 -3 -9 ] [.7381 -0.0125 3 0 ] [.97619 -0.0125 -3 -9 ] [.97619 -0.0125 3 0 ] [.2494 .12361 -6 -4.5 ] [.2494 .12361 0 4.5 ] [.2494 .24721 -6 -4.5 ] [.2494 .24721 0 4.5 ] [.2494 .37082 -6 -4.5 ] [.2494 .37082 0 4.5 ] [.2494 .49443 -6 -4.5 ] [.2494 .49443 0 4.5 ] [.2494 .61803 -12 -4.5 ] [.2494 .61803 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .02381 0 m .02381 .00625 L s [(-1)] .02381 -0.0125 0 1 Mshowa .5 0 m .5 .00625 L s [(1)] .5 -0.0125 0 1 Mshowa .7381 0 m .7381 .00625 L s [(2)] .7381 -0.0125 0 1 Mshowa .97619 0 m .97619 .00625 L s [(3)] .97619 -0.0125 0 1 Mshowa .125 Mabswid .07143 0 m .07143 .00375 L s .11905 0 m .11905 .00375 L s .16667 0 m .16667 .00375 L s .21429 0 m .21429 .00375 L s .30952 0 m .30952 .00375 L s .35714 0 m .35714 .00375 L s .40476 0 m .40476 .00375 L s .45238 0 m .45238 .00375 L s .54762 0 m .54762 .00375 L s .59524 0 m .59524 .00375 L s .64286 0 m .64286 .00375 L s .69048 0 m .69048 .00375 L s .78571 0 m .78571 .00375 L s .83333 0 m .83333 .00375 L s .88095 0 m .88095 .00375 L s .92857 0 m .92857 .00375 L s .25 Mabswid 0 0 m 1 0 L s .2619 .12361 m .26815 .12361 L s [(2)] .2494 .12361 1 0 Mshowa .2619 .24721 m .26815 .24721 L s [(4)] .2494 .24721 1 0 Mshowa .2619 .37082 m .26815 .37082 L s [(6)] .2494 .37082 1 0 Mshowa .2619 .49443 m .26815 .49443 L s [(8)] .2494 .49443 1 0 Mshowa .2619 .61803 m .26815 .61803 L s [(10)] .2494 .61803 1 0 Mshowa .125 Mabswid .2619 .0309 m .26565 .0309 L s .2619 .0618 m .26565 .0618 L s .2619 .09271 m .26565 .09271 L s .2619 .15451 m .26565 .15451 L s .2619 .18541 m .26565 .18541 L s .2619 .21631 m .26565 .21631 L s .2619 .27812 m .26565 .27812 L s .2619 .30902 m .26565 .30902 L s .2619 .33992 m .26565 .33992 L s .2619 .40172 m .26565 .40172 L s .2619 .43262 m .26565 .43262 L s .2619 .46353 m .26565 .46353 L s .2619 .52533 m .26565 .52533 L s .2619 .55623 m .26565 .55623 L s .2619 .58713 m .26565 .58713 L s .25 Mabswid .2619 0 m .2619 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 0 1 r .5 Mabswid .04836 .61803 m .0521 .53612 L .05489 .4894 L .05752 .45248 L .06244 .39698 L .06757 .35245 L .07299 .31552 L .08269 .2664 L .09312 .22896 L .10458 .19907 L .11478 .17883 L .12409 .16401 L .145 .13912 L .16409 .12302 L .18485 .10997 L .20462 .10045 L .22563 .09252 L .26338 .08207 L .28273 .07805 L .30359 .07447 L .32323 .07168 L .34473 .06916 L .38435 .06568 L .40396 .06442 L .42491 .06338 L .43515 .06297 L .44626 .0626 L .45585 .06234 L .46641 .06211 L .47136 .06203 L .47668 .06195 L .48127 .0619 L .48628 .06185 L .48921 .06184 L .49192 .06182 L .49326 .06182 L .49452 .06181 L .49597 .06181 L .49728 .06181 L .4985 .0618 L .4998 .0618 L .5009 .0618 L .50211 .0618 L .50341 .06181 L .50414 .06181 L .50481 .06181 L .50733 .06182 L .50962 .06183 L .51215 .06184 L .51727 .06188 L Mistroke .52201 .06194 L .52645 .06199 L .53645 .06217 L .54722 .06242 L .5673 .06306 L .58842 .06401 L .6082 .06517 L .62658 .0665 L .66567 .07031 L .68706 .07308 L .70722 .07624 L .74574 .08424 L .76566 .08973 L .78671 .09695 L .80633 .10543 L .82465 .11548 L .84498 .13007 L .86353 .14814 L .88259 .17435 L .89337 .19459 L .90334 .21873 L .91295 .24925 L .92198 .28785 L .92715 .31635 L .93189 .34837 L .93709 .39249 L .94258 .45379 L .94698 .51974 L .95099 .59977 L Mfstroke .95099 .59977 m .95171 .61803 L s 1 0 0 r .02869 .61803 m .03279 .56611 L .04262 .46925 L .05288 .39151 L .06244 .33527 L .0731 .28637 L .0828 .25142 L .09413 .21923 L .10458 .19573 L .12373 .16361 L .13381 .15077 L .14451 .13938 L .16404 .12299 L .18234 .11134 L .20259 .10133 L .22112 .09407 L .24171 .08757 L .26083 .08265 L .29902 .0752 L .31954 .07217 L .33815 .06988 L .35902 .06774 L .37821 .06613 L .39815 .06477 L .41676 .06375 L .43559 .06296 L .4462 .0626 L .45625 .06233 L .466 .06212 L .47114 .06203 L .4766 .06195 L .48119 .0619 L .48362 .06188 L .48624 .06186 L .48864 .06184 L .49084 .06183 L .49295 .06182 L .49516 .06181 L .4964 .06181 L .49709 .06181 L .49773 .0618 L .49891 .0618 L .50016 .0618 L .50128 .0618 L .50233 .0618 L .50347 .06181 L .50471 .06181 L .50601 .06181 L .50744 .06182 L .51003 .06183 L Mistroke .51256 .06185 L .5149 .06186 L .5238 .06196 L .52877 .06203 L .53327 .06211 L .55255 .06257 L .56341 .06292 L .57339 .06331 L .59385 .0643 L .61293 .06549 L .65095 .06871 L .67134 .071 L .68991 .07349 L .71065 .07684 L .72981 .08057 L .76819 .09051 L .78689 .09702 L .80751 .10601 L .8269 .11686 L .84777 .13231 L .86785 .15273 L .87794 .16596 L .88896 .18353 L .90861 .22628 L .9196 .25935 L .92957 .29749 L .93958 .34608 L .94879 .40285 L .95891 .4827 L .96961 .59376 L Mfstroke .96961 .59376 m .97146 .61803 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{62, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgP3oool2000001D0oooo000G0?ooo`030000003oool0oooo07D0oooo00<000000?ooo`3oool0>@3o ool00`000000oooo0?ooo`0j0?ooo`040000003oool0oooo000001@0oooo000G0?ooo`030000003o ool0oooo07D0oooo00<000000?ooo`3oool0>P3oool00`000000oooo0?ooo`0l0?ooo`030000003o ool0oooo0180oooo000?0?ooo`@00000103oool00`000000oooo0?ooo`1e0?ooo`030000003oool0 oooo03/0oooo00<000000?ooo`3oool0>@3oool2000001D0oooo000G0?ooo`030000003oool0oooo 07D0oooo00<000000?ooo`3oool0?03oool00`000000oooo0?ooo`0j0?ooo`030000003oool0oooo 0180oooo000E0?ooo`<00000M@3oool3000003/0oooo00@000000?ooo`3oool00000>@3oool01000 0000oooo0?ooo`00000D0?ooo`005`3oool00`000000oooo0?ooo`1e0?ooo`030000003oool0oooo 03X0oooo0P00000k0?ooo`8000005@3oool00?l0oooo8@3oool00?l0oooo8@3oool000d0ooooo`00 0006000000h0oooo000D0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02@3oool0 0`000000oooo0?ooo`0C0?ooo`00D@3oool00`000000oooo0?ooo`3<0?ooo`00D@3oool00`000000 oooo0?ooo`3<0?ooo`00D@3oool00`000000oooo0?ooo`3<0?ooo`00D@3oool00`000000oooo0?oo o`3<0?ooo`00D@3oool00`000000oooo0?ooo`3<0?ooo`00D@3oool00`000000oooo0?ooo`3<0?oo o`00D@3oool200000@3oool20?l000h0oooo1000 00040?ooo`030000003oool0oooo0900oooo00<0003o0?l0003oool0>@3oool003P0oooo00<0o`00 0?ooo`3oool03P3oool00`000000oooo0?ooo`050?ooo`<00000T@3oool00`000?l0o`000?l0000h 0?ooo`00=`3oool00`3o0000oooo0?ooo`0@0?ooo`030000003oool0oooo00@0oooo00<000000?oo o`3oool0T`3oool00`000?l0o`000?ooo`0f0?ooo`00=@3oool20?l001@0oooo00<000000?ooo`3o ool00`3oool00`000000oooo0?ooo`2D0?ooo`030000o`3o0000oooo03D0oooo000d0?ooo`030?l0 003oool0oooo01D0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`2E0?ooo`030000 o`3o0000oooo03@0oooo000c0?ooo`030?l0003oool0oooo01<0oooo00@000000?ooo`3oool00000 103oool00`000000oooo0?ooo`2F0?ooo`030000o`3o0000oooo03<0oooo000b0?ooo`030?l0003o ool0oooo01D0oooo0P0000050?ooo`030000003oool0oooo09P0oooo00<0o`000?ooo`3oool0<@3o ool00340oooo00<0o`000?ooo`3oool07@3oool00`000000oooo0?ooo`2I0?ooo`030?l0003oool0 oooo0300oooo000`0?ooo`030?l000000?l0oooo01h0oooo00<000000?ooo`3oool0VP3oool00`3o 0000oooo0?ooo`0_0?ooo`00;`3oool00`3o0000003o0?ooo`0O0?ooo`800000V`3oool00`000?l0 o`000?ooo`0_0?ooo`00;P3oool00`3o0000003o0?ooo`0P0?ooo`030000003oool0oooo09/0oooo 00<0o`000?ooo`3oool0;P3oool002d0oooo00<0o`000000o`3oool08@3oool00`000000oooo0?oo o`2L0?ooo`030?l0003oool0oooo02d0oooo000]0?ooo`030?l0003oool0oooo0240oooo00<00000 0?ooo`3oool0W@3oool00`3o0000oooo0?ooo`0/0?ooo`00;03oool00`3o0000003o0?ooo`0R0?oo o`030000003oool0oooo09d0oooo00<0003o0?l0003oool0;03oool002`0oooo00<0o`000?ooo`3o ool08P3oool00`000000oooo0?ooo`2N0?ooo`030?l0003oool0oooo02/0oooo000[0?ooo`030?l0 003oool0oooo02<0oooo00<000000?ooo`3oool0W`3oool00`3o0000oooo0?ooo`0Z0?ooo`00:`3o ool00`3o0000oooo0?ooo`0S0?ooo`030000003oool0oooo09l0oooo00<0003o0?l0003oool0:P3o ool002X0oooo00<0o`000?ooo`3oool0903oool200000:40oooo00<0o`000?ooo`3oool0:@3oool0 02X0oooo00<0o`000?ooo`3oool0903oool00`000000oooo0?ooo`2P0?ooo`030000o`3o0000oooo 02T0oooo000Y0?ooo`030?l0003oool0oooo02D0oooo00<000000?ooo`3oool0X@3oool00`3o0000 oooo0?ooo`0X0?ooo`00:@3oool00`3o0000oooo0?ooo`0U0?ooo`030000003oool0oooo0:40oooo 00<0003o0?l0003oool0:03oool002T0oooo00<0o`000?ooo`3oool09@3oool00`000000oooo0?oo o`2R0?ooo`030?l0003oool0oooo02L0oooo000X0?ooo`030?l0003oool0oooo02H0oooo00<00000 0?ooo`3oool0XP3oool00`000?l0o`000?ooo`0W0?ooo`00:03oool00`3o0000oooo0?ooo`0V0?oo o`030000003oool0oooo0:<0oooo00<0o`000?ooo`3oool09P3oool002L0oooo00<0o`000000o`3o ool09`3oool00`000000oooo0?ooo`2S0?ooo`030000o`3o0000oooo02H0oooo000W0?ooo`030?l0 003oool0oooo02L0oooo0P00002U0?ooo`030?l0003oool0oooo02D0oooo000V0?ooo`030?l00000 0?l0oooo02P0oooo00<000000?ooo`3oool0Y03oool00`000?l0o`000?ooo`0U0?ooo`009P3oool0 0`3o0000oooo0?ooo`0X0?ooo`030000003oool0oooo0:@0oooo00<0003o0?l0003oool09@3oool0 02D0oooo00<0o`000000o`3oool0:@3oool00`000000oooo0?ooo`2U0?ooo`030?l0003oool0oooo 02@0oooo000U0?ooo`030?l000000?l0oooo02T0oooo00<000000?ooo`3oool0Y@3oool00`000?l0 o`000?ooo`0T0?ooo`009@3oool00`3o0000oooo0?ooo`0Y0?ooo`030000003oool0oooo0:D0oooo 00<0003o0?l0003oool0903oool002@0oooo00<0o`000000o`3oool0:P3oool00`000000oooo0?oo o`2U0?ooo`030000o`3o0000oooo02@0oooo000T0?ooo`030?l000000?l0oooo02@0oooo00<00000 0?ooo`3oool00`3oool00`000000oooo0?ooo`2V0?ooo`030000o`3o0000oooo02<0oooo000S0?oo o`030?l0003oool0003o02D0oooo00<000000?ooo`3oool00`3oool300000:H0oooo00<0003o0?l0 003oool08`3oool002<0oooo00<0o`000000o`3oool08P3oool5000000@0oooo00<000000?ooo`3o ool0YP3oool00`000?l0o`000?ooo`0S0?ooo`008`3oool00`3o0000003o0?ooo`0R0?ooo`040000 003oool0oooo000000D0oooo00<000000?ooo`3oool0YP3oool00`000?l0oooo0?l0000S0?ooo`00 8P3oool00`3o0000oooo0000o`0T0?ooo`030000003oool0000000D0oooo00<000000?ooo`3oool0 Y`3oool00`000?l0o`000?ooo`0R0?ooo`008P3oool00`3o0000003o0?ooo`0U0?ooo`8000001@3o ool00`000000oooo0?ooo`2W0?ooo`030000o`3o0000oooo0280oooo000R0?ooo`030?l000000?l0 oooo02H0oooo00<000000?ooo`3oool00`3oool00`000000oooo0?ooo`2W0?ooo`030000o`3o0000 oooo0280oooo000Q0?ooo`030?l0003oool0003o02d0oooo00<000000?ooo`3oool0Y`3oool00`00 0?l0oooo0?l0000R0?ooo`008@3oool00`3o0000oooo0000o`0]0?ooo`030000003oool0oooo0:L0 oooo00<0003o0?ooo`3o00008P3oool00240oooo00<0o`000000o`3oool0;@3oool200000:T0oooo 00<0003o0?l0003oool08@3oool00200oooo00<0o`000?ooo`000?l0;P3oool00`000000oooo0?oo o`2X0?ooo`030000o`3o0000oooo0240oooo000P0?ooo`030?l0003oool0003o02h0oooo00<00000 0?ooo`3oool0Z03oool00`000?l0o`000?ooo`0Q0?ooo`00803oool00`3o0000oooo0000o`0^0?oo o`030000003oool0oooo0:P0oooo00<0003o0?ooo`3o00008@3oool00200oooo00<0o`000000o`3o ool0;P3oool00`000000oooo0?ooo`2X0?ooo`030000o`3oool0o`000240oooo000P0?ooo`030?l0 00000?l0oooo02h0oooo00<000000?ooo`3oool0Z@3oool00`000?l0o`000?ooo`0P0?ooo`007`3o ool00`3o0000oooo0000o`0_0?ooo`030000003oool0oooo0:T0oooo00<0003o0?l0003oool0803o ool001l0oooo00<0o`000?ooo`000?l0;`3oool00`000000oooo0?ooo`2Y0?ooo`030000o`3o0000 oooo0200oooo000O0?ooo`030?l000000?l0oooo02l0oooo0P00002Z0?ooo`030000o`3oool0o`00 0200oooo000O0?ooo`030?l000000?l0oooo02l0oooo00<000000?ooo`3oool0ZP3oool00`000?l0 o`000?ooo`0O0?ooo`007`3oool00`3o0000003o0?ooo`0_0?ooo`030000003oool0oooo0:X0oooo 00<0003o0?l0003oool07`3oool001l0oooo00<0o`000000o`3oool0;`3oool00`000000oooo0?oo o`2Z0?ooo`030000o`3o0000oooo01l0oooo000N0?ooo`030?l0003oool0003o0300oooo00<00000 0?ooo`3oool0ZP3oool00`000?l0oooo0?l0000O0?ooo`007P3oool00`3o0000oooo0000o`0`0?oo o`030000003oool0oooo0:X0oooo00<0003o0?ooo`3o00007`3oool001h0oooo00<0o`000?ooo`00 0?l0<03oool00`000000oooo0?ooo`2Z0?ooo`030000o`3oool0o`0001l0oooo000N0?ooo`030?l0 00000?l0oooo0300oooo00<000000?ooo`3oool0Z`3oool00`000?l0o`000?ooo`0N0?ooo`007P3o ool00`3o0000003o0?ooo`0`0?ooo`800000[03oool00`000?l0oooo0?l0000N0?ooo`007P3oool0 0`3o0000003o0?ooo`0`0?ooo`030000003oool0oooo0:/0oooo00<0003o0?ooo`3o00007P3oool0 01d0oooo00<0o`000?ooo`000?l0<@3oool00`000000oooo0?ooo`2[0?ooo`030000o`3oool0o`00 01h0oooo000M0?ooo`030?l0003oool0003o0340oooo00<000000?ooo`3oool0Z`3oool00`000?l0 oooo0?l0000N0?ooo`007@3oool00`3o0000oooo0000o`0a0?ooo`030000003oool0oooo0:/0oooo 00<0003o0?ooo`3o00007P3oool001d0oooo00<0o`000?ooo`000?l0<@3oool00`000000oooo0?oo o`2[0?ooo`030000o`3oool0o`0001h0oooo000M0?ooo`030?l0003oool0003o0340oooo00<00000 0?ooo`3oool0Z`3oool010000?l0oooo0?ooo`3o000M0?ooo`00703oool0103o0000oooo0?ooo`00 0?lZ0?ooo`8000001@3oool00`000000oooo0?ooo`2[0?ooo`040000o`3oool0oooo0?l001d0oooo 000L0?ooo`040?l0003oool0oooo0000obT0oooo00@000000?ooo`3oool00000103oool300000:/0 oooo00@0003o0?ooo`3oool0o`007@3oool001`0oooo00@0o`000?ooo`3oool0003o:@3oool01000 0000oooo0?ooo`0000040?ooo`030000003oool0oooo0:`0oooo00<0003o0?ooo`3o00007@3oool0 01`0oooo00<0o`000?ooo`000?l0:P3oool3000000D0oooo00<000000?ooo`3oool0[03oool00`00 0?l0oooo0?l0000M0?ooo`006`3oool0103o0000oooo0?ooo`000?lZ0?ooo`030000003oool0oooo 00D0oooo00<000000?ooo`3oool0[03oool00`000?l0oooo0?l0000M0?ooo`006`3oool0103o0000 oooo0?ooo`000?lZ0?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0[03oool00`00 0?l0oooo0?l0000M0?ooo`006`3oool0103o0000oooo0?ooo`000?l[0?ooo`<00000103oool00`00 0000oooo0?ooo`2/0?ooo`040000o`3oool0oooo0?l001`0oooo000K0?ooo`040?l0003oool0oooo 0000oc80oooo00<000000?ooo`3oool0[03oool010000?l0oooo0?ooo`3o000L0?ooo`006`3oool0 103o0000oooo0?ooo`000?lb0?ooo`030000003oool0oooo0:`0oooo00@0003o0?ooo`3oool0o`00 703oool001/0oooo00@0o`000?ooo`3oool0003o"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-1.3252, -0.764114, 0.0162035, 0.0624232}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["41.", "Subsubsection"], Cell[TextData[{ "The series\n\t", Cell[BoxData[ \(TraditionalForm \`sin(x)\ = \ x\ - \ x\^3\/\(3!\) + \ x\^5\/\(5!\) - x\^7\/\(7!\) + x\^9\/\(9!\) - x\^11\/\(11!\) + \ ... \)]], "\nconverges for all ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". Integrating term by term gives:\n\t ", Cell[BoxData[ \(TraditionalForm \`\(-\(cos(x)\)\) = \(\[Integral]\(sin(x)\) \[DifferentialD]x\ = \ C\ + \(1\/2\) x\^2 - \ x\^4\/\(4!\) + x\^6\/\(6!\) - x\^8\/\(8!\) + x\^10\/\(10!\) + \ ... \)\)]], "\nSince ", Cell[BoxData[ \(TraditionalForm\`cos \((0)\) = 1\)]], ", we see that ", Cell[BoxData[ \(TraditionalForm\`C = \(-1\)\)]], "and we obtain:\n\t", Cell[BoxData[ \(TraditionalForm \`cos(x)\ = \ 1\ - \ x\^2\/\(2!\) + \ x\^4\/\(4!\) - x\^6\/\(6!\) + x\^8\/\(8!\) - x\^10\/\(10!\) + \ ... \)]], "\nwhich also converges for all ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". We could also take the derivative of ", Cell[BoxData[ \(TraditionalForm\`sin(x)\)]], "\n\n If we plug in ", Cell[BoxData[ \(TraditionalForm\`2 x\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x\)]], " in the above epxression for ", Cell[BoxData[ \(TraditionalForm\`sin(x)\)]], " we get:\n\t", Cell[BoxData[ \(TraditionalForm \`sin(2 x)\ = \ 2 x\ - \ \((2 x)\)\^3\/\(3!\) + \ \((2 x)\)\^5\/\(5!\) - \((2 x)\)\^7\/\(7!\) + \((2 x)\)\^9\/\(9!\) - \((2 x)\)\^11\/\(11!\) + \ ... \)]], "\nWe know that ", Cell[BoxData[ \(TraditionalForm\`sin(2 x)\ = \ 2 \( sin(x)\) \(cos(x)\)\)]], " and if we multiply the above series for ", Cell[BoxData[ \(TraditionalForm\`sin(x)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`cos(x)\)]], " we get:\n", Cell[BoxData[ FormBox[ RowBox[{\(2 \( sin(x)\)\ \(cos(x)\)\), " ", "=", " ", RowBox[{"2", RowBox[{"(", FormBox[\(x\ - \ x\^3\/\(3!\) + \ x\^5\/\(5!\) - x\^7\/\(7!\) + x\^9\/\(9!\) - x\^11\/\(11!\) + \ ... \), "TraditionalForm"], ")"}], RowBox[{"(", RowBox[{ FormBox[ \(1\ - \ x\^2\/\(2!\) + \ x\^4\/\(4!\) - x\^6\/\(6!\) + x\^8\/\(8!\) - x\^10\/\(10!\) + \ .. \), "TraditionalForm"], StyleBox[".", FontWeight->"Bold"]}], StyleBox[")", FontWeight->"Bold"]}]}]}], TraditionalForm]]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[ 2 \((x - x\^3\/\(3!\) + x\^5\/\(5!\) - x\^7\/\(7!\) + x\^9\/\(9!\) - x\^11\/\(11!\))\) \((1 - x\^2\/\(2!\) + x\^4\/\(4!\) - x\^6\/\(6!\) + x\^8\/\(8!\) - x\^10\/\(10!\))\)]\)], "Input"], Cell[BoxData[ \(2\ x - \(4\ x\^3\)\/3 + \(4\ x\^5\)\/15 - \(8\ x\^7\)\/315 + \(4\ x\^9\)\/2835 - \(8\ x\^11\)\/155925 + \(157\ x\^13\)\/119750400 - \(19\ x\^15\)\/785862000 + \(127\ x\^17\)\/402361344000 - x\^19\/362125209600 + x\^21\/72425041920000\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Expand[\ 2 x\ - \ \((2 x)\)\^3\/\(3!\) + \ \((2 x)\)\^5\/\(5!\) - \((2 x)\)\^7\/\(7!\) + \((2 x)\)\^9\/\(9!\) - \((2 x)\)\^11\/\(11!\)]\)], "Input"], Cell[BoxData[ \(2\ x - \(4\ x\^3\)\/3 + \(4\ x\^5\)\/15 - \(8\ x\^7\)\/315 + \(4\ x\^9\)\/2835 - \(8\ x\^11\)\/155925\)], "Output"] }, Open ]], Cell[TextData[{ "Thus, multiplying out like polynomials the first six terms of the series \ for ", Cell[BoxData[ \(TraditionalForm\`2 \( sin(x)\) \(cos(x)\)\)]], " yields the first six terms series for ", Cell[BoxData[ \(TraditionalForm\`sin(2 x)\)]], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["46. (Modified)", "Subsubsection"], Cell[TextData[{ "Find the sum of the series ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] n\^3\/3\^n\)]], ".\n\nStart with ", Cell[BoxData[ \(TraditionalForm \`1\/\(1 - x\) = \[Sum]\+\(n = 0\)\%\[Infinity] x\^n\)]], ", take its derivative, ", Cell[BoxData[ \(TraditionalForm \`1\/\((1 - x)\)\^2 = \ \[Sum]\+\(n = 1\)\%\[Infinity] n\ x\^\(n - 1\)\)]], ", and multiply by ", Cell[BoxData[ \(TraditionalForm\`x\)]], " to get", Cell[BoxData[ \(TraditionalForm\`\(\(x\/\((1 - x)\)\^2 = \)\ \)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(n = 1\)\%\[Infinity] n\ x\^n\)]], ". Now take the derivative and multiply by ", Cell[BoxData[ \(TraditionalForm\`x\)]], " again: ", Cell[BoxData[ \(TraditionalForm \`x\ \(d\/\(d\ x\)\) \((x\/\((1 - x)\)\^2)\) = x \( d\/\(d\ x\)\) \(\[Sum]\+\(n = 1\)\%\[Infinity] n\ x\^n\)\)]], ". This simplifies to ", Cell[BoxData[ \(TraditionalForm \`\(x(x + 1)\)\/\((1 - x)\)\^3 = \ \[Sum]\+\(n = 1\)\%\[Infinity] n\^2\ x\^n\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\ D[x\/\((1 - x)\)\^2, x] // Simplify\)], "Input"], Cell[BoxData[ \(\(-\(\(x\ \((1 + x)\)\)\/\((\(-1\) + x)\)\^3\)\)\)], "Output"] }, Open ]], Cell[TextData[{ "Repeat these steps one more time: ", Cell[BoxData[ \(TraditionalForm \`x\ \(d\/\(d\ x\)\) \((\(x(x + 1)\)\/\((1 - x)\)\^3)\) = x \( d\/\(d\ x\)\) \(\[Sum]\+\(n = 1\)\%\[Infinity] n\^2\ x\^n\)\)]], " which simplifies to ", Cell[BoxData[ \(TraditionalForm \`\(x(x\^2 + 4 x + 1)\)\/\((1 - x)\)\^4 = \ \[Sum]\+\(n = 1\)\%\[Infinity] n\^3\ x\^n\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\ D[\(x(x + 1)\)\/\((1 - x)\)\^3, x] // Simplify\)], "Input"], Cell[BoxData[ \(\(x\ \((1 + 4\ x + x\^2)\)\)\/\((\(-1\) + x)\)\^4\)], "Output"] }, Open ]], Cell[TextData[{ "This converges absolutely for ", Cell[BoxData[ \(TraditionalForm\`\( | x | \( < 1\)\)\)]], " since the orginal geometric series converges there. Plugging in ", Cell[BoxData[ \(TraditionalForm\`x = 1\/3\)]], "gives\n\t", Cell[BoxData[ \(TraditionalForm \`\[Sum]\+\(n = 1\)\%\[Infinity] n\^3\/3\^n = \(\(\(1\/3\) \((\((1\/3)\)\^2 + 4 1\/3 + 1)\)\)\/\((1\/3 - 1)\)\^4 = 33\/8\)\)]], ".\n", StyleBox["Mathematica", FontSlant->"Italic"], " also knows how to sum this series:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\ \[Sum]\+\(n = 1\)\%\[Infinity] n\^3\/3\^n\)\)], "Input"], Cell[BoxData[ \(33\/8\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1024}, {0, 768}}, ScreenStyleEnvironment->"Working", WindowSize->{567, 592}, WindowMargins->{{Automatic, 159}, {Automatic, 35}} ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1709, 49, 67, 3, 62, "Subsection"], Cell[CellGroupData[{ Cell[1801, 56, 28, 0, 45, "Subsection"], Cell[CellGroupData[{ Cell[1854, 60, 27, 0, 42, "Subsubsection"], Cell[1884, 62, 1181, 33, 132, "Text"], Cell[CellGroupData[{ Cell[3090, 99, 108, 2, 52, "Input"], Cell[3201, 103, 238, 3, 47, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3476, 111, 178, 3, 64, "Input"], Cell[3657, 116, 20028, 510, 186, 4392, 312, "GraphicsData", "PostScript", "Graphics"], Cell[23688, 628, 130, 3, 39, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[23867, 637, 27, 0, 56, "Subsubsection"], Cell[23897, 639, 1284, 37, 201, "Text"], Cell[CellGroupData[{ Cell[25206, 680, 81, 1, 83, "Input"], Cell[25290, 683, 207, 3, 114, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25534, 691, 100, 2, 63, "Input"], Cell[25637, 695, 17378, 422, 186, 3442, 245, "GraphicsData", "PostScript", "Graphics"], Cell[43018, 1119, 130, 3, 39, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[43197, 1128, 28, 0, 56, "Subsubsection"], Cell[43228, 1130, 1258, 36, 237, "Text"], Cell[CellGroupData[{ Cell[44511, 1170, 88, 1, 83, "Input"], Cell[44602, 1173, 164, 2, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[44803, 1180, 143, 2, 75, "Input"], Cell[44949, 1184, 20226, 562, 186, 5174, 372, "GraphicsData", "PostScript", "Graphics"], Cell[65178, 1748, 130, 3, 39, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[65357, 1757, 28, 0, 56, "Subsubsection"], Cell[65388, 1759, 1471, 39, 301, "Text"], Cell[CellGroupData[{ Cell[66884, 1802, 91, 1, 83, "Input"], Cell[66978, 1805, 363, 6, 149, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[67378, 1816, 185, 3, 114, "Input"], Cell[67566, 1821, 22934, 591, 186, 5252, 368, "GraphicsData", "PostScript", "Graphics"], Cell[90503, 2414, 130, 3, 39, "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[90682, 2423, 28, 0, 56, "Subsubsection"], Cell[90713, 2425, 2695, 81, 269, "Text"], Cell[CellGroupData[{ Cell[93433, 2510, 254, 5, 51, "Input"], Cell[93690, 2517, 288, 4, 83, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[94015, 2526, 200, 4, 45, "Input"], Cell[94218, 2532, 142, 2, 45, "Output"] }, Open ]], Cell[94375, 2537, 291, 9, 50, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[94703, 2551, 39, 0, 42, "Subsubsection"], Cell[94745, 2553, 1130, 35, 117, "Text"], Cell[CellGroupData[{ Cell[95900, 2592, 71, 1, 43, "Input"], Cell[95974, 2595, 82, 1, 46, "Output"] }, Open ]], Cell[96071, 2599, 431, 12, 65, "Text"], Cell[CellGroupData[{ Cell[96527, 2615, 82, 1, 64, "Input"], Cell[96612, 2618, 83, 1, 62, "Output"] }, Open ]], Cell[96710, 2622, 566, 17, 142, "Text"], Cell[CellGroupData[{ Cell[97301, 2643, 80, 1, 79, "Input"], Cell[97384, 2646, 39, 1, 54, "Output"] }, Open ]] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)