---
tag: 0CVVKY7GY2
kind: definition
name: Convergent series
msc: [40A05]
defines: 0AF7X08PCR
uses: [0JH4MN4K1C, 0NB643K60Q, 0AZ3R52593]
sources:
  - url: https://en.wikipedia.org/wiki/Convergent_series
    locator: Definition
---

## Definition

Let $\sum a_n$ be a [[series|0JH4MN4K1C]] of real numbers with [[partial sums|0NB643K60Q]] $s_N = \sum_{n=0}^{N} a_n$.
The series is convergent if the sequence $(s_N)$ is a [[convergent sequence|0AZ3R52593]].

## Remarks

The series $\sum 1/2^n$ converges, since its partial sums $2 - 1/2^N$ tend to $2$.
The limit of the partial sums is the [[sum of the series|0K0H5E3Z5N]].
A series that does not converge is a [[divergent series|0NBRP6093V]].
Changing finitely many terms does not change whether a series converges. It can change the sum.

## Explanations

### L1 human

A series converges when its running totals approach a fixed number.
Adding more and more terms then changes the total less and less.