Cauchy–Hadamard Theorem

Cauchy–Hadamard Theorem: Consider the power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ with $a_n\in\mathbb{R}$ or $\mathbb{C}$. Define $L=\limsup_{n\to\infty}\sqrt[n]{|a_n|}\in[0,\infty].$ Then the radius of convergence $R$ is $R=\frac{1}{L},$ with the conventions $1/0=\infty$ and $1/\infty=0$. The series converges absolutely for $|x-x_0|<R$ and diverges for $|x-x_0|>R$.

This theorem is the standard quantitative description of where a power series defines a function.

Proof sketch: Apply the root test to the term $a_n(x-x_0)^n$. The $n$th root of its magnitude is $\sqrt[n]{|a_n|}\,|x-x_0|$, whose limsup is $L|x-x_0|$. The root test yields convergence when $L|x-x_0|<1$ and divergence when $L|x-x_0|>1$.