radius of convergenceRadiusOfConvergence2002-03-19 10:27:492008-06-09 18:16:47Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereTo the power series
\begin{equation}
\sum_{k=0}^{\infty}a_k(x-x_0)^k
\end{equation}
there exists a number $r\in [0,\infty]$, its \emph{radius of convergence}, such that the series converges absolutely for all (real or complex) numbers $x$ with $|x-x_0|<r$ and diverges whenever $|x-x_0|>r$. This is known as {\em Abel's theorem on power series}. (For $|x-x_0|= r$ no general statements can be made.)
The radius of convergence is given by:
\begin{equation}
r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{|a_k|}}
\end{equation}
and can also be computed as
\begin{equation}
r=\lim_{k\to\infty}\left|\frac{a_k}{a_{k+1}}\right|,
\end{equation}
if this limit exists.
It follows from the \PMlinkname{Weierstrass $M$-test}{WeierstrassMTest} that for any radius $r'$ smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius $r'$.