Bertrand criterion

of convergence of series $\sum_{n=1}^{\infty}a_n$ with positive numbers as terms

If \begin{equation} B_n=\left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n \end{equation} and if the limit (finite or infinite) \begin{equation} B = \lim_{n\to\infty}B_n \end{equation} exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand.

References

[1]	G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)