---
tag: 0660QX1VPR
kind: definition
name: Radius of convergence
msc: [30B10]
defines: 0N288P2TB5
uses: [0VP56B4EMX, 05FQ3VGAR9]
sources:
  - url: https://en.wikipedia.org/wiki/Radius_of_convergence
    locator: Definition
---

## Definition

Let $\sum a_n (x - c)^n$ be a real [[power series|0VP56B4EMX]].
Its radius of convergence is
$$R = \sup \{ r \ge 0 : \text{the sequence } (|a_n| r^n) \text{ is bounded} \},$$
an element of $[0, \infty]$.

## Remarks

The series converges absolutely for $|x - c| < R$, and it diverges for $|x - c| > R$.
At the points $x = c \pm R$, the series may converge or diverge.
Many texts define $R$ as the supremum of all $|x - c|$ where the series converges. The two definitions give the same value.
The [[Cauchy–Hadamard theorem|07E5MB1CA2]] gives a formula for $R$. Mathlib uses the bounded form, in `FormalMultilinearSeries.radius`.

## Explanations

### L1 human

A power series converges on an interval centered at $c$, and diverges outside it.
The radius of convergence is the half length of this interval.