Cauchy–Hadamard theorem

Cauchy–Hadamard theorem: For a power series $\sum_{n=0}^\infty a_n (x-x_0)^n$ (real or complex), define

Set the radius of convergence

with the conventions $1/0=\infty$ and $1/\infty=0$. Then:

• The series converges absolutely for $|x-x_0|<R$.
• The series diverges for $|x-x_0|>R$.

This result is an application of the root test to the terms $a_n(x-x_0)^n$ and is the standard way to compute the radius in practice.