Bertrand criterion

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).

#### References[edit]

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

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