Bertrand criterion

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2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

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 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).


[Fi] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
How to Cite This Entry:
Bertrand criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article