In mathematics, a Bertrand series is a series of real numbers of the form
∑ n 1 n α ( ln n ) β {\displaystyle \sum _{n}{\frac {1}{n^{\alpha }(\ln n)^{\beta }}}}
where α {\displaystyle \alpha } and β {\displaystyle \beta } are two real numbers. They are named after Joseph Bertrand. For β = 0 {\displaystyle \beta =0} , this series reduces to the series ∑ n 1 n α {\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 α > 1 {\displaystyle \alpha >1} or ( α = 1 {\displaystyle \alpha =1} and β > 1 {\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:
∑ n ≥ 2 1 n ( ln n ) 2 ≈ 2.10974 {\displaystyle \sum _{n\geq 2}{\frac {1}{n(\ln n)^{2}}}\approx 2.10974} (sequence A115563 in the OEIS);
∑ n ≥ 2 1 n ( ln n ) 3 ≈ 2.06589 {\displaystyle \sum _{n\geq 2}{\frac {1}{n(\ln n)^{3}}}\approx 2.06589} (sequence A145419 in the OEIS);
∑ n ≥ 2 1 n 2 ln n ≈ 0.60552 {\displaystyle \sum _{n\geq 2}{\frac {1}{n^{2}\ln n}}\approx 0.60552} (sequence A168218 in the OEIS).
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