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Alternating Series

Definition

A series is called an alternating series if its terms change sign alternately. One standard form, starting with a positive term, is

∑n=0∞(−1)nan=a0−a1+a2−a3+⋯\sum_{n=0}^{\infty}(-1)^n a_n=a_0-a_1+a_2-a_3+\cdots

where an>0a_n>0 for every n.n\text{.} The negative of a series of this form is also alternating.

Interpretation

An alternating series is a series whose terms switch sign at every step, a0−a1+a2−⋯a_0-a_1+a_2-\cdots with all an>0a_n>0. Consecutive terms partly cancel, so the partial sums move back and forth instead of growing monotonically.

Partial sums of the alternating harmonic series for n = 1 to 12, jumping above and below the horizontal line at ln 2 with shrinking steps.

The figure shows the partial sums sns_n of 1−12+13−⋯1-\tfrac{1}{2}+\tfrac{1}{3}-\cdots for n=1,…,12n=1,\ldots,12. They start at s1=1s_1=1 and s2=0.5s_2=0.5; after that the odd-numbered sums stay above and the even-numbered sums below the limit ln⁡2≈0.693\ln2\approx0.693, closing in from both sides, with s11≈0.737s_{11}\approx0.737 and s12≈0.653s_{12}\approx0.653.

In the definition, (−1)n(-1)^n carries the sign and ana_n the size of each term. A series that starts with a negative term, −a0+a1−⋯-a_0+a_1-\cdots, is alternating too: it is −1-1 times a series of the form above. The sign pattern alone does not give convergence: ∑(−1)n\displaystyle \sum(-1)^n is alternating and diverges. The alternating series test adds the condition that ana_n decreases to 00.

An alternating series can converge although ∑an\displaystyle \sum a_n diverges: the alternating harmonic series in the figure converges, while the harmonic series diverges. A geometric series with a positive first term and a negative ratio is alternating, which is why, for ∣q∣<1|q|<1, its partial sums approach the limit from alternate sides.

Key Results

  • Alternation of signs may produce convergence through cancellation, even when the corresponding series of absolute values diverges.

Examples

  • ∑n=0∞(−1)n1n+1\displaystyle \sum_{n=0}^\infty (-1)^n \frac{1}{n+1}
  • 1−12+13−14+⋯\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Prerequisites

Sources

Symbols: infinity, sum

Last updated: 27 October 2026 11:52