Raabe criterion

on the convergence of a series of numbers

A series $\sum_{n=1}^{\infty}a_n$ converges if for sufficiently large $n$ the inequality \begin{equation} R_n = n\left(\frac{a_n}{a_{n+1}}-1\right)\geq r>1 \end{equation} is fulfilled. If $R_n\leq 1$ from some $n$ onwards, then the series diverges.

Proved by J. Raabe

References

[a1]	K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990)