# Alternating series

A series whose terms switch sign. If the terms shrink steadily towards 0, it converges. The error is smaller than the first term you leave out.

Maths (university) · Part 07: Sequences and series · Card 7 of 9 · Last updated 2026-10-19

## Analogy

Walking two steps forward and one step back, with steps that shrink: you settle at a point.

## Example

**1 - 1/2 + 1/3 - 1/4 + ...**

Terms shrink to 0, signs alternate.
So it converges (to ln 2, about 0.693).
S_4 = 1 - 0.5 + 0.333 - 0.25 = 0.583.
Error less than 1/5 = 0.2.

## In practice

Lets you estimate sums with a guaranteed error bound.

## Practice

### Flashcards

- **When does an alternating series converge?** When its terms shrink steadily to 0.

## Questions people ask

### What is the difference between absolute and conditional convergence?

A series converges absolutely if the sum of the absolute values converges. It converges conditionally if the series converges only because of sign cancellation. The alternating harmonic series converges, but 1 + 1/2 + 1/3 and so on does not, so it is conditional.

### How accurate is a partial sum of an alternating series?

The error is smaller than the first term you left out, and the true sum sits between consecutive partial sums. With 1 - 1/2 + 1/3 - 1/4, the next term is 1/5, so the answer is within 0.2 of 0.583.

## Related concepts

- [Partial sum](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-2)
- [Geometric series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-3)
- [Divergence test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-4)
- [p-series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-5)

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