# Abel’s limit theorem

Suppose that $\sum a_{n}x^{n}$ has a radius of convergence $r$ and that $\sum a_{n}r^{n}$ is convergent. Then

 $\lim_{x\to r^{-}}\sum a_{n}x^{n}=\sum a_{n}r^{n}=\sum(\lim_{x\to r^{-}}a_{n}x^% {n})$
Title Abel’s limit theorem AbelsLimitTheorem 2013-03-22 12:57:49 2013-03-22 12:57:49 CWoo (3771) CWoo (3771) 5 CWoo (3771) Theorem msc 40A30 PowerSeries AbelsMultiplicationRuleForSeries AbelSummability NielsHenrikAbel