Alternating series test

Statement

Let $(b_n)$ be a decreasing real sequence with $b_n \to 0$ as $n \to \infty$. Then the alternating series $\sum_{n=0}^{\infty} (-1)^n b_n$ converges. Moreover, its sum $S$ and its partial sums $s_N = \sum_{n=0}^{N} (-1)^n b_n$ satisfy $|S - s_N| \le b_{N+1}$ for each $N$.

Remarks

The even partial sums $s_0, s_2, s_4, \dots$ decrease, and the odd partial sums $s_1, s_3, \dots$ increase. Each odd partial sum is below each even one, so both sequences converge by the monotone convergence theorem. Their limits agree, since $s_{2k} - s_{2k+1} = b_{2k+1} \to 0$. The hypothesis that $(b_n)$ decreases is needed. Without it, an alternating series with terms tending to $0$ can diverge. Mathlib states the convergence for an antitone sequence, in Antitone.tendsto_alternating_series_of_tendsto_zero.

History

Leibniz discussed the criterion in an unpublished work of 1676. For this reason it is also called the Leibniz test.

Explanations

L1 human

Each step goes in the other direction and is shorter than the step before. So the partial sums swing back and forth with smaller and smaller swings, and they settle down.