View source for Raabe criterion

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''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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,   "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964)  (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>