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Homeradius of convergence

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# radius of convergence

To the power series

$\sum_{{k=0}}^{{\infty}}a_{k}(x-x_{0})^{k}$ | (1) |

there exists a number $r\in[0,\infty]$, its *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 Abel’s theorem on power series. (For $|x-x_{0}|=r$ no general statements can be made.)

The radius of convergence is given by:

$r=\liminf_{{k\to\infty}}\frac{1}{\sqrt[k]{|a_{k}|}}$ | (2) |

and can also be computed as

$r=\lim_{{k\to\infty}}\left|\frac{a_{k}}{a_{{k+1}}}\right|,$ | (3) |

if this limit exists.

It follows from the Weierstrass $M$-test that for any radius $r^{{\prime}}$ smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius $r^{{\prime}}$.

Related:

ExampleOfAnalyticContinuation, NielsHenrikAbel

Synonym:

Abel's theorem on power series

Type of Math Object:

Theorem

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

40A30*no label found*30B10

*no label found*

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## Attached Articles

## Corrections

Abel's theorem by pahio ✓

uniform convergence by rspuzio ✓

radius of convergence by ogu ✓

nomenclature, definition by ogu ✓

images by ogu ✘

uniform convergence by rspuzio ✓

radius of convergence by ogu ✓

nomenclature, definition by ogu ✓

images by ogu ✘