Root test

Statement

Let $(a_n)$ be a real sequence and let $C = \limsup_{n \to \infty} |a_n|^{1/n}$, the limit superior, with the value $+\infty$ allowed. If $C < 1$, then the series $\sum a_n$ converges absolutely. If $C > 1$, then the series $\sum a_n$ diverges.

Remarks

If $C < 1$, choose $q$ with $C < q < 1$. Then $|a_n| \le q^n$ for large $n$, and the comparison test with a geometric series applies. If $C > 1$, then $|a_n| > 1$ for infinitely many $n$, so the terms do not tend to $0$. For $C = 1$ the test gives no answer. The root test is stronger than the ratio test: when the ratio limit exists, the root limit exists and has the same value.

Explanations

L1 human

The $n$-th root of $|a_n|$ measures how fast the terms shrink, compared with a geometric sequence $q^n$. If this rate is below $1$, the terms shrink like a convergent geometric series.