Radius of convergence

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

A power series converges for $|x - a| < R$ and diverges for $|x - a| > R$ . Surprisingly, $R$ is the distance from $a$ to the nearest singularity in the complex plane: the series of $\frac{1}{1 + x^2}$ at 0 has $R = 1$ because of the poles at $\pm i$ , although the function is perfectly smooth on $\R$ .

Formulas

$\frac{1}{R} = \limsup_{k\to\infty}|c_k|^{1/k}$
Cauchy–Hadamard: $\frac{1}{R} = \lim_{k\to\infty}\left|\frac{c_{k+1}}{c_k}\right|$
when this limit exists (ratio test)

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.