---
tag: FJKV01J9F1
kind: concept
name: Convergence tests
msc: [40A05]
sources:
  - url: https://en.wikipedia.org/wiki/Convergence_tests
same_as:
  wikidata: Q3979448
  wikipedia: Convergence tests
---

## Summary

A convergence test is a rule that decides, from a property of the terms, whether a [[series|0JH4MN4K1C]] converges or diverges.
Most series have no closed form for their sum, so such rules are the main way to settle convergence.
Standard examples are the [[comparison test|04Z5G0THN6]], the [[ratio test|0QNY6ZBXPN]], the [[root test|05H4DDDYBZ]] and the [[integral test|019QZ6RRVH]].
The exact definition is in [[convergence test|MA9EMC488R]].

## Explanations

### L1 human

A convergence test is a quick check that tells you whether an infinite sum settles down to a number.
You look at the terms, not at the sum itself.
For example, each term of $\sum 1/2^n$ is half of the term before, and the ratio test then says the series converges.
A test can also stay silent: for $\sum 1/n$ and $\sum 1/n^2$ the ratio test gives no answer.