Divergence part of the ratio test

Statement

Let $(a_n)$ be a real sequence with $a_n \ne 0$ for all $n$. Assume that $|a_{n+1}| / |a_n| \to L$ as $n \to \infty$, for a real number $L$ with $L > 1$. Then the series $\sum a_n$ diverges.

Remarks

For large $n$ we have $|a_{n+1}| > |a_n|$. So $|a_n|$ is eventually increasing and positive, and $a_n$ does not tend to $0$. The term test gives divergence. The same proof works when $|a_{n+1}| / |a_n| \to +\infty$. For example, $\sum 2^n / n$ diverges, since the ratio is $2n/(n+1) \to 2$.

Explanations

L1 human

If each term is eventually bigger than the term before, the terms cannot shrink to $0$. Then the series cannot converge.