Alternating Series

Summary

An alternating series has terms that change sign. Leibniz’s test gives a simple convergence criterion, and the error after $N$ terms is at most the next unused term’s magnitude.

Prerequisites

Sequences, Series Sums by Partial Sums

Definition

Typical forms:

\sum_{n = 1}^{\infty} (- 1)^{n + 1} b_{n} or \sum_{n = 1}^{\infty} (- 1)^{n} b_{n},

with $b_{n} \geq 0$ .

Theorem (Leibniz / alternating series test)

If $(b_{n})$ is eventually monotone decreasing and $lim_{n \to \infty} b_{n} = 0$ , then the alternating series converges.

Remainder

If the hypotheses hold for all $n \geq 1$ (or from the first unused index onward),

| S - S_{N} | \leq b_{N + 1} .

Worked Example

The alternating harmonic series

\sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots

has $b_{n} = 1 / n ↓ 0$ , so it converges (to $\ln 2$ ). It does not converge absolutely.

For $\sum_{n = 1}^{\infty} (- 1)^{n + 1} / 2^{n}$ , $b_{n} = 2^{- n}$ decreases to $0$ , so the series converges (in fact absolutely, since it is geometric with ratio $- 1 / 2$ ).

Common Mistakes

Applying the test when $b_{n}$ is not eventually decreasing.
Confusing conditional convergence with absolute convergence.
Off-by-one index errors in remainder statements.

Connections

Absolute Convergence, Estimating Series Sums, Harmonic Series

References

The alternating series test is in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 5.5, https://openstax.org/details/books/calculus-volume-2 ↩︎