Let $\sum_{n=0}^\infty a_n$ be an infinite series with $a_n\neq 0$ for all $n \ge N$, where $N\in\mathbb N$ is some index (i.e. all but the first $N$ sequence members $a_n$ must be different from $0$). Furthermore, assume that there exists a positive $q$ with $0 < q < 1$ such that $$\left|\frac{a_{n+1}}{a_n}\right|\le q$$ for all $n \ge N.$ Then $\sum_{n=0}^\infty a_n$ is an absolutely convergent series. But if for an index $N\in\mathbb N$ we have $$\left|\frac{a_{n+1}}{a_n}\right|\ge 1\text{ for all }n \ge N,$$ then the series is divergent.
Proofs: 1
