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

$$\underset{x\to r^{-}}{lim}\sum a_n{x}^n=\sum a_n{r}^n=\sum (\underset{x\to r^{-}}{lim}{a}_n{x}^n)$$

Title	Abel’s limit theorem
Canonical name	AbelsLimitTheorem
Date of creation	2013-03-22 12:57:49
Last modified on	2013-03-22 12:57:49
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	5
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 40A30
Related topic	PowerSeries
Related topic	AbelsMultiplicationRuleForSeries
Related topic	AbelSummability
Related topic	NielsHenrikAbel