Root test

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

What is it?

If $\limsup |a_k|^{1/k} = L$ : $L < 1$ converges, $L > 1$ diverges. Slightly stronger than the ratio test; it gives the radius of convergence of power series (Cauchy–Hadamard).

Formulas

\limsup_{k\to\infty}|a_k|^{1/k} < 1 \implies \sum |a_k| < \infty

What depends on it

Radius of convergence

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