Bertrand series

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

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

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

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

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

One has the following approximate values:

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

References

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