# Bertrand criterion

2010 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 $$B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\,$$ 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

 [Fi] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
How to Cite This Entry:
Bertrand criterion. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=30917
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article