---
tag: 0Y123X48GA
kind: statement
role: theorem
name: Absolutely convergent series converge
msc: [40A05]
uses: [0W3K3QPEWT, 06PVXMY0TH]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_convergence
    locator: Relation to convergence
---

## Statement

Let $(a_n)$ be a real sequence.
If the series $\sum a_n$ [[converges absolutely|0W3K3QPEWT]], then $\sum a_n$ converges.
Moreover, $\left| \sum_{n=0}^{\infty} a_n \right| \le \sum_{n=0}^{\infty} |a_n|$.

## Remarks

The proof uses the [[Cauchy criterion for series|06PVXMY0TH]]. For $n > m$ the triangle inequality gives $\left| \sum_{k=m+1}^{n} a_k \right| \le \sum_{k=m+1}^{n} |a_k|$.
The converse fails: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but $\sum 1/n$ diverges.
The result holds in every complete normed space. Mathlib states it as `Summable.of_norm`.

## Explanations

### L1 human

Cancellation between terms of opposite signs can only make partial sums smaller in absolute value.
So if the series converges without any cancellation, it also converges with it.