Editing Radius of convergence (section)

== Convergence on the boundary ==
If the power series is expanded around the point ''a'' and the radius of convergence is {{math|''r''}}, then the set of all points {{math|''z''}} such that {{math|1={{mabs|''z'' − ''a''}} = ''r''}} is a [[circle]] called the ''boundary'' of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily converge absolutely.

Example 1: The power series for the function {{math|1=''f''(''z'') = 1/(1 − ''z'')}}, expanded around {{math|1=''z'' = 0}}, which is simply
:<math> \sum_{n=0}^\infty z^n,</math> 
has radius of convergence 1 and diverges at every point on the boundary.

Example 2: The power series for {{math|1=''g''(''z'') = −ln(1 − ''z'')}}, expanded around {{math|1=''z'' = 0}}, which is
:<math> \sum_{n=1}^\infty \frac{1}{n} z^n,</math> 
has radius of convergence 1, and diverges for {{math|1=''z'' = 1}}  but converges for all other points on the boundary. The function {{math|''f''(''z'')}} of Example 1 is the [[derivative]] of {{math|''g''(''z'')}}.

Example 3: The power series
:<math> \sum_{n=1}^\infty \frac 1 {n^2} z^n </math> 
has radius of convergence 1 and converges everywhere on the boundary absolutely. If {{math|''h''}} is the function represented by this series on the unit disk, then the derivative  of ''h''(''z'') is equal to ''g''(''z'')/''z'' with ''g'' of Example 2. It turns out that {{math|''h''(''z'')}} is the [[dilogarithm]] function.

Example 4: The power series
:<math>\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for } n = \lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n,</math>
has radius of convergence 1 and converges [[uniform convergence|uniformly]] on the entire boundary {{math|1={{mabs|''z''}} = 1}}, but does not [[Absolute convergence|converge absolutely]] on the boundary.<ref>{{cite journal |url=https://eudml.org/doc/215384|title=O szeregu potęgowym, który jest zbieżny na całem swem kole zbieżności jednostajnie, ale nie bezwzględnie|journal=Prace Matematyczno-Fizyczne|year=1918|volume=29|issue=1|pages=263–266|last1=Sierpiński|first1=W.|author-link=Wacław Sierpiński}}</ref>