Alternating series

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 is alternating if its terms are nonzero and their signs alternate.

Remarks

Some texts allow $b_n \ge 0$. The alternating harmonic series $1 - 1/2 + 1/3 - \cdots$ is the standard example. The alternating series test 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.