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Bertrand criterion

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Bertrand criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=29179
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article