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  Exam 1  &\huge Tmath 126  &\normalsize Spring 2011\\
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\begin{enumerate}

\item $[8]$ TRUE/FALSE: Identify a statement as True in each of the following cases if the 
statement is \emph{always} true and provide a brief justification.  Otherwise, identify it as false and provide a counterexample. 

\begin{enumerate}
%Quiz 2 T/F
\item An infinite sum of nonzero terms will never converge to a finite number.
\vfill
%practice test #3
\item The series, $\displaystyle{ \sum_{n=1}^{\infty} \frac{2}{n} }$ converges.
\vfill
%lecture 4/6
\item For all real numbers $x$, $\displaystyle{e^{ix}=\cos(x)+i\sin(x)}$, where $i=\sqrt{-1}$.
\vfill
\end{enumerate}


Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit.

%WebHW2 #3
\item $[4]$ Write the following sum using the sigma notation:
\[ - \frac{1}{4} + \frac{2}{9} - \frac{3}{16} + \frac{4}{25} \]
\vskip 1in


\pagebreak

\item $[8]$ Compute the following if possible.  

%practice exam #3  %WebHW2 #10
$\displaystyle{ \sum_{n=0}^{\infty} \frac{(2)^n}{n!} }$ \hspace{2in} $\displaystyle{ \sum_{n = 1}^{\infty} \frac{(2e)^n}{6^{n-1}} }$

\vskip 2.5in
%(Quiz 2 \#4) 
\item  Let $\{a_n\}_{n=1}^\infty$ be a sequence such that the $n^{\text{th}}$ partial sum of a series  is $\displaystyle{s_n=\frac{n-n^2}{3n^2-e}}$.
\begin{enumerate}
\item $[3]$ Find $\displaystyle{\sum_{n=1}^\infty a_n}$, if it exists.  Justify your answer.
\vfill
\item $[3]$ Find $a_n$. You do \emph{not} need to simplify the expression.
\vfill
\item $[2]$ Find $\displaystyle{\lim_{n\rightarrow \infty}a_n}$, if it exists. Justify your answer.
\vfill
\end{enumerate}


\pagebreak
%lecture 4/4
\item $[3]$ For what $x$ values with the power series $\displaystyle{\sum_{n=1}^\infty (x-4)^n}$ converge?
\vskip 1.5in

%practice test #5
\item  Let $\displaystyle{f(x) =\ln (3x)}$.
\begin{enumerate}
\item $[6]$ Find the second order Taylor polynomial $T_2 (x)$ based at $b =\frac{1}{3}$.
\vfill
%WebHW4 #1-3
\item $[4]$ Bound the error $| f(x) - T_2 (x) |$ on the interval $\left[\frac{1}{6},\frac{1}{2}\right]$.
\vfill
\end{enumerate}


\pagebreak

%lecture 4/11
\item  Consider the function $e^{-x^2}$.
\begin{enumerate}
\item $[3]$ Find the Taylor series for the function $e^{-x^2}$.
\vskip 1.5in
\item $[3]$ Find $\displaystyle{\int_0^x e^{-t^2} \, dt}$.
\vskip 2in
\end{enumerate}

\vspace{-1in}\hspace{1in}\includegraphics[height=5in]{BlankTrig}

\vspace{-4.5in} \item  Consider the sequence: $\{a_1, 2\sin(a_1), 2\sin(2\sin(a_1)), 2\sin(2\sin(2\sin(a_1))), ... \}$
\begin{enumerate}
\item $[1]$ The sequence \\
above is an iterative \\
sequence where \\
$a_n=f(a_{n-1})$.  \\
Find $f(x)$.
\vskip 1in
\item $[1]$ Draw the graph \\
of $f$ on the axes \\
provided.\\
\item $[1]$ If $a_1=\frac{\pi}{4}$, identify what the sequence above converges to on the graph.  
\vskip .3in
\item $[2]$ For what values of $a_1$ between $-2\pi$ and $\pi$ will the above sequence converge to a negative finite number?
\end{enumerate}
\end{enumerate}

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