Ratio test

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

What is it?

If $|a_{k+1}/a_k| \to L$ : $L < 1$ converges, $L > 1$ diverges, $L = 1$ says nothing. It compares the series with a geometric one, and it is the natural test for series with factorials.

Formulas

\lim_{k\to\infty}\left|\frac{a_{k+1}}{a_k}\right| = L < 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.