---
tag: 0K0H5E3Z5N
kind: definition
name: Sum of a series
msc: [40A05]
defines: 0HH4Y7HSRQ
uses: [0CVVKY7GY2, 0NB643K60Q]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
    locator: Sum of a series
---

## Definition

Let $\sum a_n$ be a [[convergent series|0CVVKY7GY2]] of real numbers with partial sums $s_N = \sum_{n=0}^{N} a_n$.
The sum of the series is the real number
$$\sum_{n=0}^{\infty} a_n = \lim_{N \to \infty} s_N.$$

## Remarks

The sum is well defined because a sequence has at most one limit.
The sum is linear: $\sum (a_n + b_n) = \sum a_n + \sum b_n$ and $\sum c\, a_n = c \sum a_n$ when $\sum a_n$ and $\sum b_n$ converge. This follows from the [[algebra of limits|07XBM3A9J5]].
For example, $\sum_{n=0}^{\infty} 1/2^n = 2$.

## Explanations

### L1 human

The sum of a series is the number that its running totals approach.
For $1 + 1/2 + 1/4 + \cdots$, the totals $1, 1.5, 1.75, 1.875, \dots$ approach $2$, so the sum is $2$.