---
tag: 0VVY4H92G2
kind: statement
role: proposition
name: Term test
aliases: [nth-term-test, divergence-test]
msc: [40A05]
uses: [0CVVKY7GY2, 0NB643K60Q, 0RAM4E07NR]
sources:
  - url: https://en.wikipedia.org/wiki/Nth-term_test
    locator: Usage
  - url: https://en.wikipedia.org/wiki/Nth-term_test
    locator: Proofs
same_as:
  wikidata: Q2576932
  wikipedia: Nth-term test
---

## Statement

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers.
Then $a_n \to 0$ as $n \to \infty$.
Equivalently, if $(a_n)$ does not converge to $0$, then the series $\sum a_n$ diverges.

## Remarks

For the proof, let $s$ be the sum. Then $a_n = s_n - s_{n-1} \to s - s = 0$.
The converse is false. The terms of the [[harmonic series|0CXTMAY5PX]] tend to $0$, and the series diverges.
For example, $\sum (-1)^n$ and $\sum n/(n+1)$ diverge by this test.

## Explanations

### L1 human

If you keep adding terms that do not get small, the total cannot settle down.
So small terms are necessary for convergence, but they are not enough.