Abel's Limit Theorem

analysis

Definition

Abel's Limit Theorem

Let $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ be a power series with radius of convergence $R=1$. If the series converges at $x=1$, then:

$$\underset{x\to 1^{-}}{\lim }f(x)=\sum _{n=0}^{\infty }{a}_n$$

More generally, if the series has radius of convergence $R$ and converges at $x=R$, then ${\lim }_{x\to R^{-}}f(x)={\sum }_{n=0}^{\infty }{a}_n{R}^n$. The theorem ensures that the function defined by the power series is continuous from the left at the boundary of its interval of convergence.

Significance

Abel’s theorem is a key result in the study of power series because the radius of convergence only guarantees convergence (and continuity) inside the open interval $(-R,R)$. This theorem provides a bridge to the boundary $x=R$.

Example: Gregory’s Series

The power series for $\arctan (x)$ is ${\sum }_{n=0}^{\infty }\frac{(-1{)}^n}{2n+1}{x}^{2n+1}$, which has $R=1$. At $x=1$, the series is the alternating Leibniz series for $\pi /4$. Abel’s theorem allows us to conclude:

$$\lim \_x\to 1^{-}\arctan (x)=\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$$