---
tag: 07E5MB1CA2
kind: statement
role: theorem
name: Cauchy–Hadamard theorem
aliases: [cauchy-hadamard-formula]
msc: [30B10]
uses: [0660QX1VPR, 0P8BA7T7S5, 05H4DDDYBZ]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy–Hadamard_theorem
    locator: Theorem for one complex variable
same_as:
  wikidata: Q859122
  wikipedia: Cauchy–Hadamard theorem
---

## Statement

Let $\sum a_n (x - c)^n$ be a real power series with [[radius of convergence|0660QX1VPR]] $R$.
Then
$$\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n},$$
with the conventions $1/0 = \infty$ and $1/\infty = 0$.

## Remarks

The proof applies the [[root test|05H4DDDYBZ]] to the series $\sum a_n (x - c)^n$. The [[limit superior|0P8BA7T7S5]] of $|a_n (x - c)^n|^{1/n}$ is $|x - c| \limsup |a_n|^{1/n}$.
For example, $\sum n x^n$ has $R = 1$, since $n^{1/n} \to 1$.
Mathlib states it as `FormalMultilinearSeries.radius_inv_eq_limsup`.

## Explanations

### L1 human

The coefficients $a_n$ grow or shrink roughly like $\rho^n$, where $\rho$ is the limit superior of $|a_n|^{1/n}$.
Then $a_n x^n$ behaves like $(\rho x)^n$, which is small exactly when $|x| < 1/\rho$.