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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Representations of Functions as Power Series">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Representations of Functions as Power Series}

\subsubsection{Differentiation and Integration of Power Series\label%
{differentiation and integration of power series}}

\begin{theorem}
If the power series $\sum c_{n}\left( x-a\right) ^{n}$ has \hyperref{radius
of convergence}{}{}{cal09_08.tex#radius of convergence} $R>0$, then the
function $f$ defined by 
\[
f\left( x\right) =c_{0}+c_{1}\left( x-a\right) +c_{2}\left( x-a\right)
^{2}+...=\sum\limits_{n=0}^{\infty }c_{n}\left( x-a\right) ^{n}
\]
is differentiable (and therefore continuous) on the interval $\left(
a-R,a+R\right) $ and

\begin{enumerate}
\item  $f\prime \left( x\right) =c_{1}+2c_{2}\left( x-a\right) +3c_{3}\left(
x-a\right) ^{2}+...=\sum\limits_{n=1}^{\infty }nc_{n}\left( x-a\right) ^{n-1}
$

\item  $\int f\left( x\right) dx=C+c_{0}\left( x-a\right) +c_{1}\frac{\left(
x-a\right) ^{2}}{2}+c_{2}\frac{\left( x-a\right) ^{3}}{3}+...=C+\sum%
\limits_{n=0}^{\infty }c_{n}\frac{\left( x-a\right) ^{n+1}}{n+1}$
\end{enumerate}
\end{theorem}

The radii of convergence of the power series in equations 1 and 2 are both $%
R $.

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