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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 81 "Animation of the Convergence of
 Partial Sums of a Fourier Series to the Function " }}{PARA 258 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Here is a Maple proced
ure which lets you animate the convergence of the partial sums of the \+
Fourier series of a function " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT 
-1 332 " to the function.  The procedure requires four arguments: the \+
function, the independent variable and its range (for the plot), and a
 positive integer (which partial sum will be the last one you use).  T
he procedure will calculate the Fourier series on an interval whose le
ngth is the length of the range of the independent variable." }}{PARA 
257 "" 0 "" {TEXT -1 1 " " }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 630 "fourier_picture := proc( func, xrange::name=range, n
::posint)\n   local x, a, b, L, k, j, p, q, partsum;\n   a:=lhs( rhs(x
range));\n   b:=rhs( rhs(xrange));\n   L:=b-a;\n   x:=2*Pi*lhs(xrange)
/L;\n   partsum:=1/L * evalf( Int( func, xrange));\n   for k from 1 to
 n do\n     partsum:=partsum \n        + 2/L *evalf( Int(func*sin(k*x)
, xrange)) *sin(k*x) \n        + 2/L *evalf( Int(func*cos(k*x), xrange
)) *cos(k*x);\n     q[k] := plot( partsum, xrange, color=blue,args[4..
nargs]);\n   od;\n   q:= plots[display]( [seq( q[k], k=1..n)], inseque
nce=true);\n   p:=plot( func, xrange, color=red, args[4..nargs]);\n   \+
plots[display]( [q,p] );\n   end:\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 19 "Here I use it with " }{XPPEDIT 18 0 "f(
x) = abs(x);" "6#/-%\"fG6#%\"xG-%$absG6#F'" }{TEXT -1 133 ".  I'll use
 [-1,1] as the range. We've already seen that the partial sums converg
e very fast, so I'll go up to the fifth partial sum." }{TEXT -1 156 " \+
 To animate, you have to click with the left mouse button on the plot.
  Then the video controls will appear as a toolbar, right under the st
andard toolbar." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "fourier_picture(abs(x),x=-1..1,5);" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Next I'll use \+
" }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF'" }{TEXT -1 112 " agai
n on the interval [-1,1].  We've seen that convergence takes longer, s
o I'll go up to the 20th partial sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fourier_picture(x,x=-1..
1,20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 15 "Here I'll take " }{XPPEDIT 18 0 "f(x
) = exp(x);" "6#/-%\"fG6#%\"xG-%$expG6#F'" }{TEXT -1 64 " on the inter
val [0,10].  Again, I'll take the 20th partial sum." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "fourier_pi
cture(exp(x), x=0..10, 20);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{MARK "19" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 
0 1 2 33 1 1 }