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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Power Series">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
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\section{Power Series\textbf{\label{power series}}}

A \textbf{power series} is a series of the form 
\[
\sum\limits_{n=0}^{\infty }c_{n}x^{n}=c_{0}+c_{1}x+c_{2}x^{2}+... 
\]
where $x$ is a variable and the $c_{n}$'s are called the \textbf{coefficients%
} of the series. The sum of the series is a function 
\[
f\left( x\right) =c_{0}+c_{1}x+c_{2}x^{2}+... 
\]
whose domain is the set of all $x$ for which the series converges. A series
of the form 
\[
\sum\limits_{n=0}^{\infty }c_{n}\left( x-a\right) ^{n}=c_{0}+c_{1}\left(
x-a\right) +c_{2}\left( x-a\right) ^{2}+... 
\]
is called a \textbf{power series in }$\left( x-a\right) $ or a \textbf{power
series centered at }$a$ or a \textbf{power series about }$a$. The series
always converges when $x=a$ since all terms are $0$ for $n\geq 1$.

\begin{example}
The \textbf{Bessel function\label{Bessel function}} of order $0$ is defined
by 
\[
J_{0}\left( x\right) =\sum\limits_{n=0}^{\infty }\frac{\left( -1\right)
^{n}x^{2n}}{2^{2n}\left( n!\right) ^{2}}
\]
The series converges for all values of $x$. That is, the domain of the
Bessel function is $\UNICODE{0x211d} $.
\end{example}

\begin{theorem}
For a given power series $\sum\limits_{n=0}^{\infty }c_{n}\left( x-a\right)
^{n}$, there are three possibilities:
\end{theorem}

\begin{enumerate}
\item  The series converges only when $x=a$.

\item  The series converges for all $x$.

\item  There is a positive number $R$ such that the series converges when $%
\left| x-a\right| <R$ and diverges when $\left| x-a\right| >R$.
\end{enumerate}

The number $R$ in case 3 is called the \textbf{radius of convergence\label%
{radius of convergence}} of the power series. In case 1, the radius of
convergence is $0$, and in case 2 it is $\infty $. The \textbf{interval of
convergence\label{interval of convergence}} of a power series is the
interval that contains all values of $x$ for which the series converges. In
case 1, the interval of convergence is $a$. In case 2, the interval of
convergence is $\left( -\infty ,\infty \right) $. In case 3, there are four
possibilities for the interval of convergence: $\left( a-R,a+R\right) $, $%
\left( a-R,a+R\right] $, $\left[ a-R,a+R\right) $, $\left[ a-R,a+R\right] $%
.\bigskip 

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Bessel functions, notation}{}{}{%
../../Help/DM5-6.tex#Special Functions}}

{\small \hyperref{Maclaurin series}{}{}{../../Help/DM7-7.tex#Power series}}

{\small \hyperref{Plotting Bessel functions}{}{}{%
../../Help/DM10-3.tex#Bessel functions}}

{\small \hyperref{Power series}{}{}{../../Help/DM7-7.tex#Power series}}

{\small \hyperref{Taylor series}{}{}{../../Help/DM7-7.tex#Taylor series}}

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