Theorem

Abel's Theorem (Power Series)

Analysis

Abel's Theorem describes the behavior of a power series at the boundary of its interval or disk of convergence: if the series converges at a boundary point, then the function it defines is continuous up to that point, and its value there equals the sum of the series. Proved by Niels Henrik Abel, it justifies taking limits of a power series at points where ordinary convergence arguments alone are not sufficient.

Facts

Statement

Abel's theorem relates the limit of a power series, taken as its argument approaches the boundary of its interval of convergence, to the sum of the series' coefficients. 1

Proof Year

1826 1

Classification

Statement Form

Identity or Equation 1

Connections

Sources

1. Abel's Theorem, Power Series (Wikipedia)

Wikimedia Foundation

Lead section, opening definition
Abel's theorem for power series relates a limit of a power series to the sum of its coefficients.
Lead section, sentence naming Abel and 1826
It is named after Norwegian mathematician Niels Henrik Abel, who proved it in 1826.

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Abel's Theorem (Power Series)

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