Editing Power series (section)

=== Behavior near the boundary ===

The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:

# ''Divergence while the sum extends to an analytic function'': <math display="inline">\sum_{n=0}^{\infty}z^n</math> has radius of convergence equal to <math>1</math> and diverges at every point of <math>|z|=1</math>. Nevertheless, the sum in <math>|z|<1</math> is <math display="inline">\frac{1}{1-z}</math>, which is analytic at every point of the plane except for <math>z=1</math>.
# ''Convergent at some points divergent at others'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n}</math> has radius of convergence <math>1</math>. It converges for <math>z=-1</math>, while it diverges for <math>z=1</math>.
# ''Absolute convergence at every point of the boundary'': <math display="inline">\sum_{n=1}^{\infty}\frac{z^n}{n^2}</math> has radius of convergence <math>1</math>, while it converges absolutely, and uniformly, at every point of <math>|z|=1</math> due to [[Weierstrass M-test]] applied with the [[Harmonic series (mathematics)#p-series|hyper-harmonic convergent series]] <math display="inline">\sum_{n=1}^{\infty}\frac{1}{n^2}</math>.
# ''Convergent on the closure of the disc of convergence but not continuous sum'': [[Wacław Sierpiński|Sierpiński]] gave an example<ref>{{cite journal|author=Wacław Sierpiński|title=Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue. (French)|journal=Rendiconti del Circolo Matematico di Palermo| url=https://zbmath.org/?q=an:46.1466.03|year=1916|volume=41|publisher=Palermo Rend.|pages=187–190 | doi=10.1007/BF03018294 |jfm=46.1466.03 | s2cid=121218640| author-link=Wacław Sierpiński}}</ref> of a power series with radius of convergence <math>1</math>, convergent at all points with <math>|z|=1</math>, but the sum is an unbounded function and, in particular, discontinuous. A sufficient condition for one-sided continuity at a boundary point is given by [[Abel's theorem]].