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%  Representing Functions with Power Series        %
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\problemgrouptitle{Representing Functions with Power Series}

\begin{problemblock}{Find a power series representation for $f(x)$ and specify  the interval of convergence.}

\item $\displaystyle f(x)=\frac 1{1+x^2}$
\answer{
$\displaystyle \sum_{n=0}^{\infty}(-1)^n x^{2n},\    I=(-1, 1)$
}

\item $\displaystyle f(x)=\frac x{5-4x}$
\answer{
$\displaystyle \sum_{n=0}^{\infty}\frac {4^n}{5^{n+1}} x^{n+1},\    I=\left(-\frac 54, \frac 54\right)$
}

\item $\displaystyle f(x)=\frac x{2+x^2}$
\answer{
$\displaystyle \sum_{n=0}^{\infty}(-1)^n \frac 1{2^{n+1}}x^{2n+1},\    I=\left (-\sqrt 2, \sqrt 2 \right)$
}

\item $\displaystyle f(x)=\frac 1{(2+3x)^2}$
\answer{
$\displaystyle -\frac 16 \sum_{n=0}^{\infty}(-1)^n \frac {n  3^n}{2^n}x^{n-1},\    I=\left(-\frac 23, \frac 23 \right)$
}

\item $\displaystyle f(x)=\ln (3+2x)$
\answer{
$\displaystyle  \sum_{n=0}^{\infty}(-1)^n \frac { 2^{n+1}}{(n+1)  3^{n+1}}x^{n+1}+\ln 3,\    I=\left(-\frac 32, \frac 32 \right)$
}

\end{problemblock}

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