Bertrand series

In mathematics, a Bertrand series is a series of real numbers of the form

$\sum _n\frac{1}{n^{\alpha }(\ln n{)}^{\beta }}$

where $\alpha$ and $\beta$ are two real numbers. They are named after Joseph Bertrand.

For $\beta =0$, this series reduces to the series $\sum _n\frac{1}{n^{\alpha }}$ defining the Riemann zeta function.

One can show a necessary and sufficient condition of convergence for such series: it converges if and only if $\alpha >1$ or ($\alpha =1$ and $\beta >1$). It can be proved by using the integral test for convergence or the Cauchy condensation test.

One has the following approximate values:

• $\sum _{n\ge 2}\frac{1}{n(\ln n{)}^2}\approx 2.10974$ (sequence A115563 in the OEIS);
• $\sum _{n\ge 2}\frac{1}{n(\ln n{)}^3}\approx 2.06589$ (sequence A145419 in the OEIS);
• $\sum _{n\ge 2}\frac{1}{n^2\ln n}\approx 0.60552$ (sequence A168218 in the OEIS).