---
tag: 017BR1M98J
kind: definition
name: Alternating series
msc: [40A05]
defines: 0QED0GPYKY
uses: [0JH4MN4K1C]
sources:
  - url: https://en.wikipedia.org/wiki/Alternating_series
    locator: lead section
---

## Definition

Let $(b_n)$ be a real sequence with $b_n > 0$ for all $n$.
The series
$$\sum_{n=0}^{\infty} (-1)^n b_n = b_0 - b_1 + b_2 - b_3 + \cdots$$
is an alternating series.
More generally, a [[series|0JH4MN4K1C]] is alternating if its terms are nonzero and their signs alternate.

## Remarks

Some texts allow $b_n \ge 0$.
The [[alternating harmonic series|0JQ5Y3WWVE]] $1 - 1/2 + 1/3 - \cdots$ is the standard example.
The [[alternating series test|0PK8QH1W1V]] gives a simple condition for convergence.

## Explanations

### L1 human

In an alternating series, the signs of the terms go plus, minus, plus, minus.
The partial sums then move up and down, like a pendulum.