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\title{Math 1b. Representations of Functions by Power Series}

\date{Spring 2006}

\begin{document}

\maketitle

\begin{center}
\begin{tabular}{|c|c|c|} \hline
Function & Series & Interval of Convergence \\ \hline
$e^x$ &  $\displaystyle 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ & $(-\infty, \infty)$ \\  \hline
$\sin x$ &  $\displaystyle x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ & $(-\infty, \infty)$ \\ \hline
$\cos x$ & $\displaystyle 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ & $(-\infty, \infty)$ \\ \hline
$\displaystyle\frac{1}{1-x}$ & $1 + x + x^2 + x^3 + \cdots$ & $(-1,1)$ \\ \hline
$\displaystyle\frac{1}{1+x}$ & $1 - x + x^2 - x^3 + \cdots$ & $(-1,1)$ \\ \hline
$\displaystyle\frac{1}{1 + x^2}$ & $1 - x^2 + x^4 - x^6 + \cdots$ & $(-1,1)$ \\ \hline
$\arctan x$ & $\displaystyle x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$ & $[-1, 1]$\\ \hline
$\ln (x + 1)$ & $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$ & $(-1,1)$ \\ \hline
\end{tabular}
\end{center}

\begin{enumerate}

\item
Find a power series representation for $\displaystyle f(x) = \frac{x^2}{1+ x}$.


\clearpage


\item
Find a power series representation for $\displaystyle f(x) = \frac{1}{(1+ x)^2}$.

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\item
Find a power series representation for $\displaystyle f(x) = \frac{1}{2 + x}$ and determine the interval of convergence.

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\item
Find a power series representation for $\displaystyle f(x) = x^2 \cos x$.

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\item
Find a power series representation for $\displaystyle f(x) = \cos(x^2)$.

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\item
Show that $\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots = \ln 2$.

\vfill




\end{enumerate}


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