---
tag: 0NBRP6093V
kind: definition
name: Divergent series
msc: [40A05]
defines: 0JWMJSTBSG
uses: [0CVVKY7GY2]
sources:
  - url: https://en.wikipedia.org/wiki/Divergent_series
    locator: lead section
---

## Definition

A [[series|0JH4MN4K1C]] $\sum a_n$ of real numbers is divergent if it is not a [[convergent series|0CVVKY7GY2]].
Thus its sequence of partial sums has no limit in $\mathbb{R}$.

## Remarks

The series $1 + 1 + 1 + \cdots$ diverges, and its partial sums tend to $+\infty$.
The series $1 - 1 + 1 - 1 + \cdots$ diverges, and its partial sums oscillate between $1$ and $0$.
The [[harmonic series|0CXTMAY5PX]] diverges, although its terms tend to $0$.
Summation methods, such as Cesàro summation, give values to some divergent series.

## Explanations

### L1 human

A divergent series has no sum in the usual sense.
Its running totals grow without bound or keep jumping around.