# Divergence test

If the terms of a series do not tend to 0, the series diverges. Terms tending to 0 is necessary, but does not guarantee convergence.

Maths (university) · Part 07: Sequences and series · Card 4 of 9 · Last updated 2026-10-19

## Analogy

If you keep adding chunks that never get small, the pile must keep growing.

## Example

**Series of n / (n + 1)**

Terms: 1/2, 2/3, 3/4, ...
Limit of n/(n+1) is 1, not 0.
Terms stay close to 1.
So the series diverges.

## In practice

A fast first check before any harder test.

## Practice

### Flashcards

- **What does the divergence test say?** If the terms do not tend to 0, the series diverges.

## Questions people ask

### If the terms go to zero, must the series converge?

No. The harmonic series 1 + 1/2 + 1/3 + and so on has terms tending to zero, yet it diverges. The test can only prove divergence. When the terms do tend to zero, you must try a different test to decide.

### Why must the terms of a convergent series go to zero?

If the running total settles on a number, consecutive totals get very close together. Their difference is the next term, so that term must get tiny. Terms that stay near 1 would keep pushing the total up by about 1 every step.

## 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)
- [p-series](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-5)
- [Ratio test](https://studyvisually.com/learn/science/maths-university/maths-university-part-7/maths-university-part-7-concept-6)

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