# Bertrand criterion

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).