Abel's Summations Theorem

Theorem (Abel’s Summations)

If ${\sum }_{n=0}^{\infty }{c}_n{x}^n$ converges at (WLOG, let the center $x=0$) the end point of the interval of convergence, then the power series can be extended continuously to the end point.

For example, if the interval of convergence is $(-R,R)$ and converges at $x=R$, then the function $f(x)={\sum }_{n=0}^{\infty }{c}_n{x}^n$ can be extended continuously to include that end point. This means

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

This allows us to describe the behavior of the interval of convergence at its endpoints.

Theorem (Double Sequences)

$$\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{a}_{m,n}=\sum _{n=1}^{\infty }\sum _{n=1}^{\infty }{a}_{m,n}$$

Where $\{a_{m,n}\}$ is a double sequence.

Theorem (Expanding a Power Series)

A power series in its open interval of convergence can be expanded as a power series everywhere. So if

$$\sum _{n=0}^{\infty }{c}_n{x}^n$$

converges on $(-R,R)$, then for any $a\in (-R,R)$, we can re-expand it as a new Power Series,

$$\sum _{n=0}^{\infty }{d}_n(x-a{)}^n$$

This is the same as

$$\sum c_n[(x-a)+a{]}^n$$