---
tag: 0AZ3R52593
kind: definition
name: Convergent sequence
msc: [40A05]
defines: 0W4CN9E2W5
uses: [0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Real numbers
---

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is convergent if it converges to some real number $L$.
Here convergence is in the sense of the [[limit of a sequence|0XKWSSY8P2]].

## Remarks

The sequences $1/(n+1)$ and $(n+1)/(n+2)$ are convergent, with limits $0$ and $1$.
The constant sequence $a_n = c$ converges to $c$.
A sequence that is not convergent is a [[divergent sequence|01HR2AH0BX]].

## Explanations

### L1 human

A convergent sequence is a sequence that has a limit.
Its terms settle down near one real number.