Abel's theorem

Statement

Let $(a_n)$ be a real sequence such that the series $\sum_{n=0}^{\infty} a_n$ converges. Let $G(x) = \sum_{n=0}^{\infty} a_n x^n$ for $-1 < x < 1$. Then $G(x) \to \sum_{n=0}^{\infty} a_n$ as $x \to 1$ from the left.

Remarks

The power series converges on $(-1, 1)$, since its radius of convergence is at least $1$. The theorem gives the sum of the alternating harmonic series. Apply it to the Mercator series for $\ln(1 + x)$. The converse fails. For $a_n = (-1)^n$, $G(x) = 1/(1 + x) \to 1/2$, but $\sum (-1)^n$ diverges. Mathlib states it as Real.tendsto_tsum_powerSeries_nhdsWithin_lt.

Explanations

L1 human

A power series that converges at the endpoint $x = 1$ is continuous there from the left. So its value at $1$ is the limit of its values inside the interval.