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\begin{center}
Math 132: Calculus for Physical and Life Sciences 2\\ 
Problem Set 9 \\ 
Due Friday, April 18, 2008, at the beginning of class.
\end{center}
\vspace{10pt}

General Directions: You must show all work for credit on these problems. 

\begin{enumerate}
\item Determine whether each of the following series converges
absolutely, converges conditionally, or diverges, using some
combination of the Alternating Series Test, the Ratio Test, and
the Integral Test. 

\begin{enumerate}
\item[(a)]  ${\displaystyle \sum_{n=0}^\infty \frac{(-1)^n}{e^n}}$
$\qquad\qquad\qquad\qquad$ (b) ${\displaystyle \sum_{n=1}^\infty \frac{(-1)^n (2n+1)}{4n-3}}$
\item[(c)]  ${\displaystyle \sum_{n=1}^\infty \frac{1}{n 7^n}}$
$\qquad\qquad\qquad\qquad\quad$ (d) ${\displaystyle \sum_{n=2}^\infty \frac{(-1)^{n-1}}{\ln(n)}}$
\item[(e)]  ${\displaystyle \sum_{n=1}^\infty \frac{n^n}{n!}}$
\item[](Hint for (e): Try the Ratio Test on this one because of the powers and factorials.
The limit you will need to compute is a $1^\infty$ indeterminate form.
These are covered in Section 4.5 of the text if you need to review.)
\end{enumerate}

\item For each of the following power series, 
\begin{enumerate}
\item[(i)] Determine the radius of convergence $R$ using the Ratio Test.
\item[(ii)] If $0< R < \infty$, determine whether the series converges at $a \pm R$.
\item[(iii)] Based on (i) and (ii), give the interval of convergence.
\end{enumerate}

\begin{enumerate}
\item[(a)]  ${\displaystyle \sum_{n=1}^\infty \frac{(x - 2)^n}{n}}$
$\qquad\qquad\qquad\qquad$ (b)  ${\displaystyle \sum_{n=0}^\infty \frac{2^n x^n}{n!}}$
\item[(c)]  ${\displaystyle \sum_{n=0}^\infty n! (x-4)^n }$
\end{enumerate}

\item Using the definition, find the
Taylor series of each of the following functions $f(x)$
at the indicated $a$.  Write the series using summation notation and also
give the first four nonzero terms.

\begin{itemize}
\item[(a)]  $f(x) = e^{2x}$, $a = 0$.
\item[(b)]  $f(x) = \cos(x)$, $a = \pi/2$.
\end{itemize}

\item 
\begin{enumerate}
\item[(a)] Starting from the geometric series expansion for 
${\displaystyle f(x) = \frac{1}{1+x}}$, determine a power series 
(with $a = 0$) representing the 
function $\ln(1 + x)$.
\item[(b)] What is the radius of convergence of your series from part (a)?
\item[(c)] Compute the Taylor series of $\ln(1 + x)$ at $a = 0$.  How
do your answers to parts (a) and (c) compare?
\end{enumerate}

\item 
\begin{enumerate}
\item[(a)] Starting from the Taylor series for $f(t) = \sin(t)$, find a 
power series representation for the function 
\[
Si(x) = \int_0^x \frac{\sin(t)}{t}\, dt.
\]
\item[(b)]  Find a power series representation for the function 
\[
F(x) = \int_0^x \frac{1}{1 - t^2}\, dt.
\]
\end{enumerate}

\item  Compute the requested Taylor polynomials.
\begin{enumerate}
\item[(a)] The Taylor polynomial of degree $n = 3$ for
$f(x) = \tan(x)$ at $a = 0$.
\item[(b)] The Taylor polynomial of degree $n = 6$ for
$f(x) = \cos(4x)$ at $a = 0$.  
\item[(c)] The Taylor polynomial of degree $n = 4$ for
$f(x) = {\root 3\of {x - 1}}$ at $a = 2$.  
\end{enumerate}

\item
\begin{enumerate}
\item[(a)] Compute the Taylor polynomial of degree $n = 4$ for
$f(x) = \sqrt{1 + x}$ at $a = 0$.  
\item[(b)]  Use your polynomial to approximate $\sqrt{1.1}$.  (What
$x$ should you use to do this?)
\item[(c)]  What is the error of your approximation, taking a
calculator value for $\sqrt{1.1}$ as the exact value?
\end{enumerate}

\item  Use Taylor series, not L'Hopital's Rule, to evaluate
each of the following limits.
\begin{enumerate}
\item[(a)]  ${\displaystyle \lim_{x\to 0} \frac{1 - \cos(x)}{x(e^x - 1)}}$.
\item[(b)]  ${\displaystyle \lim_{x\to 0} \frac{e^x - e^{-x} - 2x}{x - \tan^{-1}(x)}}$.
\end{enumerate}
\end{enumerate}
\vfill\eject
\noindent
\emph{Selected Answers}
\vskip 10pt
1.  
\begin{enumerate}
\item[(a)] Converges absolutely.
\item[(d)] Converges conditionally (i.e. the series itself converges, but the series
of absolute values does not).
\item[(e)] Diverges.
\end{enumerate}

\vskip 10pt
2. 
\begin{enumerate}
\item[(a)] (iii) Interval of convergence is $[1,3)$ (that is all real $x$ with $1 \le x < 3$).
\item[(c)] (iii) Interval of convergence is $\{4\}$ (that is the series converges only for $x = 4$).
\end{enumerate}

\vskip 10pt
3. (b) ${\displaystyle \sum_{k=0}^\infty (-1)^{k+1} \frac{(x - \pi/2)^{2k+1}}{(2k+1)!} }$

\vskip 10pt
4. (a) ${\displaystyle \ln(1 + x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n}}$
for all $x$ with $|x| < 1$.  

\vskip 10pt
5. (b) ${\displaystyle \sum_{n = 0}^\infty \frac{x^{2n+1}}{2n+1}}$.

\vskip 10pt
6. (b) ${\displaystyle p_6(x) =  1 - 8x^2 + \frac{32x^4}{3} - \frac{256 x^6}{45}.}$

\vskip 10pt
8. (b) $1$.
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