\magnification1200
%by J. Palmieri
\newcount\problemcount
\problemcount=1
\def\problemA{\number\problemcount a.\  }
\def\problemB{\number\problemcount b.\  %
\global\advance\problemcount by 1}

\def\problem{\number\problemcount.\  %
\global\advance\problemcount by 1}
\def\correct{}
\def\wrong{}
\def\answer#1#2#3#4#5{\vskip 10true pt
(a)\ #1\hfill(b)\ #2\hfill(c)\ #3\hfill(d)\ #4\hfill(e)\ #5}

\def\ll#1{%
\hskip10pt#1\quad$\underline{\hskip 1true in}$\vskip10pt%
}
\newtoks\xxfott
\xxfott=\expandafter{\the\footline}
\footline{}
\long\def\boxit#1{\setbox0=\hbox{#1}%
\ifvmode\vskip -\dp0\else\lower\dp0\fi %
\vbox{\hrule\hbox{\vrule\box0 \vrule}\hrule}}

\def\arc{{\rm arc}\hskip-1pt}
\def\frac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
\def\dd#1\wrt#2{\frac{d(#1)}{d{#2}}}
\def\DD#1\wrt#2{\frac{d#1}{d{#2}}}
\def\Int{\displaystyle\int}
\def\Frac#1#2{\frac{\displaystyle #1}{\displaystyle #2}}
\def\sfrac#1#2{\frac{\scriptstyle #1}{\scriptstyle #2}}
\long\def\hide#1\endhide{}
\font\big=cmr12 at 16true pt
\font\bigbold=cmbx12 at 16true pt
\parindent=0pt
\def\mcab{\hskip 10pt\boxit{\lower .25 true in%
\vbox to .5true in{\vfill\hsize=.5true in\hfill}}}
\settabs\+  01\quad&184\quad&Nieuwland \quad
&11:00 - 11:50\quad&Instructor\cr
%%%%%%%%%%%%%%%%%%%%
\centerline{\bigbold Math 126}
\centerline{\bigbold Fall 1997}
\centerline{\bigbold Test 3}
\centerline{\bigbold Dec. 2, 1997}
\vskip 20true pt
{\big
{{\parindent=1true in
\itemitem{$\bullet$}
This test runs for one hour.
\itemitem{$\bullet$}
All work is to be your own.
\itemitem{$\bullet$}
Calculators are not to be used.
}}

\vbox to 1true in{
\vskip .5true in
\noindent{\bigbold Name:}\quad$\underline{\hskip 3true in}$\par
Professor:
$\underline{\hbox to 1true in{\hfill Wong\hfill}}$\hfill Lab. Section \#
$\underline{\hskip .5true in}$
\vfill}

\centerline{\bigbold Grading on this examination.}
\centerline{\bigbold READ CAREFULLY BEFORE BEGINNING.}
\vskip 10true pt
The two multiple choice questions are either correct or wrong.
Please write the letter of the correct answer in the box provided.
The remaining questions will require you to demonstrate 
knowledge of a correct procedure for arriving at the answer.
Please circle the answer you want us to consider.
The correct answer is not enough for full credit, 
you must indicate how you arrived at it; in particular, please specify
which convergence tests you use.  Please mark out any work on a
problem that you do not want us to consider.
}
%\vskip .25true in
\vbox to 0pt{\hbox{%
\vtop{\hsize=4true in{\big
Please write your name and your Lab.
section \# at the top of this sheet.
Write your name at the top of each subsequent sheet.
If the name of your Professor is not the name at the top of this page,
raise your hand.}
\vskip 20 true pt
Your Section \# choices are:
{{\obeylines
\tabalign \#  &    Room    & &        Time   &     Instructor\cr
\tabalign  01&184& Nieuwland&11:00 - 11:50 &Jager\cr
\tabalign  02 &129& DeBartolo&12:55 -  1:45&Jager\cr
\tabalign  03&184& Nieuwland &3:55 -  4:45&Vassiliev\cr
\tabalign  04 &140& DeBartolo &2:00 -  2:50&Jager\cr
\tabalign  05 &205 &Cushing &2:00 -  2:50&Vassiliev\cr
\tabalign  06&318 &DeBartolo&9:30 - 10:20&Vassiliev\cr
}}
}
\hskip 10true pt
\raise 20pt\vbox{\hskip50true pt {\bf Score:}\par\noindent
\hbox{\boxit{\vtop{\hsize=2true in
\vskip 14pt
\ll{1.}\ll{2.}\ll{3.}\ll{4.}\ll{5.}\ll{6.}\ll{7.}\ll{8.}
$\underline{\hskip\hsize}$\vskip10pt%
\ll{Total: }%
}}}
}}
\vss}

\vfill\eject
\headline{Name:$\underline{\hskip 4true in}$\hfill}
\footline={\the\xxfott}
\null\vskip 4true pt
\problem(5pt)
The Maclaurin series for $\sin (2x^{2})$ starts off as:
\mcab

(a) {$2x^{2} - \frac{4x^{6}}{3} + \frac{4x^{10}}{15} - \dots$}
\vskip10pt
(b) {$1 - \frac{2x^{4}}{2!}  + \frac{2x^{8}}{4!} - \dots$}
\vskip10pt
(c) {$2x^{2} - \frac{2x^{6}}{3!} + \frac{2x^{10}}{5!} - \dots$\ \ \ \ }
\vskip10pt
(d) {$1 - 2x^{4} + \frac{2x^{8}}{3} - \dots$}
\vskip10pt
(e) {none of the above}

%\answer
%{$2x^{2} - \frac{4x^{6}}{3} + \frac{4x^{10}}{15} - \dots$}
%{$1 - \frac{2x^{4}}{2!}  + \frac{2x^{8}}{4!} - \dots$}
%{$2x^{2} - \frac{2x^{6}}{3!} + \frac{2x^{10}}{5!} - \dots$\ \ \ \ }
%{$1 - 2x^{4} + \frac{2x^{8}}{3} - \dots$}
%{none of the above}
\vfill
\problem(5pt)
$\displaystyle \sum_{n=0}^{\infty} \frac{3}{(n+2)(n+3)} =\ ?$
\mcab\answer
{$0$}
{$\Frac{1}{2}$}
{$1$}
{$\Frac{3}{2}$}
{diverges}
\vfill\eject

\problem(10pt) 
For each of the following series, does it converge or diverge?  If it
converges, identify the sum.  Give reasons for your answers.
\vskip10pt
(a) $\displaystyle \sum_{n=0}^{\infty} (-1)^{n+1} \pi^{-n}$.

\vfill

(b) $\displaystyle \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$.

\vfill\eject

\problem(20pt)
Determine whether each of the following series converges absolutely, 
converges conditionally, or diverges.  Give reasons for your answers.
\vskip10pt
(a) $\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^{n}}{3n-2}$ 
\vfill
(b) $\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^{n} n}{3n^{3}-2}$
\vfill\eject
(c) $\displaystyle \sum_{n=1}^{\infty} \left(\frac{3}{n}\right)^{n}$
\vfill
(d) $\displaystyle \sum_{n=0}^{\infty} \frac{(-1)^{n}n^{2}}{\sqrt{n!}\ {}}$
\vfill\eject

\problem(15pt)
What is the interval of convergence of $\displaystyle 
\sum_{n=1}^{\infty} \frac{(2x)^{n}}{n^{2}}$?

\vfill\eject

\problem(15pt)
Find the Taylor series for $f(x) = x^{2} + e^{x}$ about $x=1$.

\vfill\eject

\problem(15pt)
Express $\displaystyle \int_{0}^{1}e^{-x^{2}} dx$ as an infinite 
series.  How accurate an estimate do you get from the first four 
terms of this series?  (You may express your answer as a decimal or a
fraction.)

\vfill\eject

\problem(15pt)
Use the binomial theorem to evaluate
$$
\lim_{x\rightarrow 0} \frac{(1+x^{2})^{3/4} - (1+ \frac{3}{4}x^{2})}
{x^{4}}.
$$


\end



