View source for Bertrand criterion

{{MSC|40A05}}
{{TEX|done}}

''for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers''

A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by [[Joseph Bertrand|J. Bertrand]]. Assume that the limit
\begin{equation}
B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\, 
\end{equation}
exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence cannot be decided, as it is witnessed by the examples
\[
\sum_{n\geq 2} \frac{1}{n \log n}
\]
(which diverges) and
\[
\sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\, 
\]
(which converges).

====References====
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|valign="top"|{{Ref|Fi}}|| G.M. Fichtenholz,   "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)
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