Abel's Limit Theorem

analysis

Definition

Abel's Limit Theorem

Let $f(x)={\sum }_{n=0}^{\infty }{a}_n(x-x_0{)}^n$ be the sum of a power series with radius of convergence $R$. Then the limit ${\lim }_{x\to R}f(x)$ exists, and it satisfies

$$\underset{x\to R}{\lim }f(x)=\sum _{n=0}^{\infty }{a}_n{R}^n$$

Of course, not all functions can be expanded into a power series. A necessary condition that such a function $f$ must fulfil is obviously that $f$ is infinitely often continuously differentiable.