Abel's Limit Theorem

analysis

Definition

Abel's Limit Theorem

Let be a power series with radius of convergence . If the series converges at , then:

More generally, if the series has radius of convergence and converges at , then . 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 . This theorem provides a bridge to the boundary .

Example: Gregory’s Series

The power series for is , which has . At , the series is the alternating Leibniz series for . Abel’s theorem allows us to conclude: