---
tag: 01HR2AH0BX
kind: definition
name: Divergent sequence
msc: [40A05]
defines: 0P3M4ZY9X8
uses: [0AZ3R52593]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Real numbers
---

## Definition

A [[real sequence|0CTV783HYT]] $(a_n)$ is divergent if it is not a [[convergent sequence|0AZ3R52593]].
Thus, for each real number $L$, the sequence $(a_n)$ does not converge to $L$.

## Remarks

The sequence $a_n = (-1)^n$ is divergent, and it is bounded.
The sequence $a_n = n$ is divergent, and it [[tends to infinity|09AZCV6GMC]].
Some texts say that a sequence that tends to $+\infty$ diverges to $+\infty$. It is still divergent in the sense of this definition.

## Explanations

### L1 human

A divergent sequence has no limit.
Its terms can grow without bound, as in $1, 2, 3, \dots$, or they can jump between values, as in $1, -1, 1, -1, \dots$.