---
tag: 0JH4MN4K1C
kind: definition
name: Series
aliases: [infinite-series]
msc: [40A05]
defines: 0NM626FH6Q
uses: [0CTV783HYT]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
    locator: Definition
---

## Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a [[real sequence|0CTV783HYT]].
The series with terms $a_n$ is the sequence $(s_N)_{N \in \mathbb{N}}$ given by
$$s_N = a_0 + a_1 + \cdots + a_N = \sum_{n=0}^{N} a_n.$$
The notation for the series is $\sum_{n=0}^{\infty} a_n$, or $\sum a_n$.

## Remarks

The number $s_N$ is a [[partial sum|0NB643K60Q]] of the series.
The symbol $\sum_{n=0}^{\infty} a_n$ also denotes the [[sum of the series|0K0H5E3Z5N]], when the series is convergent. The context shows which meaning is intended.
A series can start at another index, for example $\sum_{n=1}^{\infty} 1/n$.

## Explanations

### L1 human

A series is an infinite sum $a_0 + a_1 + a_2 + \cdots$.
We cannot add infinitely many numbers directly, so we add the first terms and watch what happens as we add more.