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\problemgrouptitle{Maclaurin Series}

\begin{problemblock}{Find the Maclaurin series for $f(x)$ and state the radius of  convergence.}

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

\item $\displaystyle f(x)=x^2 e^{2x}$
\answer{
$\displaystyle \sum_{n=0}^{\infty} \frac {2^n}{n!}x^{n+2},\    R=\infty$
}

\item $\displaystyle f(x)=\cos^2 x$
\answer{
$\displaystyle\frac 12+\frac 12 \sum_{n=0}^{\infty}(-1)^n \frac {2^{2n}}{(2n)!}x^{2n},\    R=\infty$
}

\item $\displaystyle f(x)=x^2 \sin 4x$
\answer{
$\displaystyle \sum_{n=0}^{\infty}(-1)^n \frac {4^{2n+1}}{(2n+1)!}x^{2n+3},\    R=\infty$
}

\item $\displaystyle f(x)=\cos \left(x^3\right)$
\answer{
$\displaystyle \sum_{n=0}^{\infty}(-1)^n \frac {1}{(2n)!}x^{6n},\    R=\infty$
}

\end{problemblock}

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