Editing Power series (section)

==Radius of convergence==

A power series <math display="inline"> \sum_{n=0}^\infty a_n(x-c)^n</math> is [[convergent series|convergent]] for some values of the variable {{math|''x''}}, which will always include {{math|1=''x'' = ''c''}} since <math>(x-c)^0 = 1</math> and the sum of the series is thus <math>a_0</math> for {{math|1=''x'' = ''c''}}. The series may [[divergent series|diverge]] for other values of {{mvar|x}}, possibly all of them. If {{math|''c''}} is not the only point of convergence, then there is always a number {{math|''r''}} with {{math|0 < ''r'' ≤ ∞}} such that the series converges whenever {{math|{{abs|''x'' – ''c''}} < ''r''}} and diverges whenever {{math|{{abs|''x'' – ''c''}} > ''r''}}.  The number {{math|''r''}} is called the [[radius of convergence]] of the power series; in general it is given as
<math display="block">r = \liminf_{n\to\infty} \left|a_n\right|^{-\frac{1}{n}}</math>
or, equivalently,
<math display="block">r^{-1} = \limsup_{n\to\infty} \left|a_n\right|^\frac{1}{n}.</math>
This is the [[Cauchy–Hadamard theorem]]; see [[limit superior and limit inferior]] for an explanation of the notation. The relation
<math display="block">r^{-1} = \lim_{n\to\infty}\left|{a_{n+1}\over a_n}\right|</math>
is also satisfied, if this limit exists.

The set of the [[complex number]]s such that {{math|{{abs|''x'' – ''c''}} < ''r''}} is called the [[disc of convergence]] of the series. The series [[absolute convergence|converges absolutely]] inside its disc of convergence and it [[uniform convergence|converges uniformly]] on every [[compact space|compact]] [[subset]] of the disc of convergence.

For {{math|1={{abs|''x'' – ''c''}} = ''r''}}, there is no general statement on the convergence of the series. However, [[Abel's theorem]] states that if the series is convergent for some value {{mvar|z}} such that {{math|1={{abs|''z'' – ''c''}} = ''r''}}, then the sum of the series for {{math|1=''x'' = ''z''}} is the limit of the sum of the series for {{math|1=''x'' = ''c'' + ''t'' (''z'' – ''c'')}} where {{mvar|t}} is a real variable less than {{val|1}} that tends to {{val|1}}.