Cauchy–Hadamard theorem

Statement

Let $\sum a_n (x - c)^n$ be a real power series with radius of convergence $R$. 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 to the series $\sum a_n (x - c)^n$. The limit superior 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 $|x| < 1/\rho$.