Abel's test

Statement

Let $\sum a_n$ be a convergent real series. Let $(b_n)$ be a monotone and bounded real sequence. Then the series $\sum a_n b_n$ converges.

Remarks

The sequence $(b_n)$ converges to some $b$, by the monotone convergence theorem. Then $\sum a_n b_n = \sum a_n (b_n - b) + b \sum a_n$. The first series converges by Dirichlet's test, since $b_n - b$ is monotone with limit $0$. The hypotheses on $\sum a_n$ are stronger than in Dirichlet's test, and those on $(b_n)$ are weaker.

History

Niels Henrik Abel proved the test in 1826.

Explanations

L1 human

Multiplying the terms of a convergent series by a bounded monotone sequence of weights keeps convergence. The weights settle down to a limit, so they cannot disturb the series too much.