---
tag: 0NB643K60Q
kind: definition
name: Partial sum
msc: [40A05]
defines: 09RJ859N7A
uses: [0JH4MN4K1C]
sources:
  - url: https://en.wikipedia.org/wiki/Series_(mathematics)
    locator: Partial sum of a series
---

## Definition

Let $\sum_{n=0}^{\infty} a_n$ be a [[series|0JH4MN4K1C]] of real numbers and let $N$ be a natural number.
The $N$-th partial sum of the series is the finite sum
$$s_N = \sum_{n=0}^{N} a_n.$$

## Remarks

The partial sums satisfy $s_0 = a_0$ and $s_N = s_{N-1} + a_N$ for $N \ge 1$.
The terms are the differences of the partial sums: $a_N = s_N - s_{N-1}$ for $N \ge 1$.
For the series $\sum 1/2^n$, the partial sums are $1, 3/2, 7/4, 15/8, \dots$, and $s_N = 2 - 1/2^N$.

## Explanations

### L1 human

The partial sums are the running totals of the series.
You add one more term at each step and record the total.