This convergence test is due to Joseph Raabe (1801 - 1859).
Let $\sum_{n=0}^\infty a_n$ be an infinite series with such that for a given positive number $\beta > 1$ there exists an index $N$ such that $a_n\neq 0$ and $$\left|\frac{a_{n+1}}{a_n}\right|\le 1-\frac{\beta}{n}$$ for all $n\ge N,$ then the infinite series $\sum_{n=0}^\infty a_n$ is absolutely convergent. But if there is an index $N\in\mathbb N$ such that $$\left|\frac{a_{n+1}}{a_n}\right|\ge 1-\frac{1}{n}\quad\forall n\ge N,$$ then $\sum_{n=0}^\infty a_n$ is divergent.
Proofs: 1
