The series limit of sum_k 1/[k log k (log log k)^2]

TL;DR

This paper accurately evaluates the sum of a slowly converging series involving logarithmic terms by combining integral approximation and bias correction, providing precise numerical results for specific parameter values.

Contribution

It introduces a method to evaluate complex logarithmic series by replacing the tail with an integral and correcting biases, achieving high-precision results.

Findings

Sum for a=2: 38.4067680928

Method applicable to other a values like 3 and 4

Provides a systematic approach for series with slow convergence

Abstract

The slowly converging series sum_{k=3}^infinity 1/[k * log k * (log log k)^a] is evaluated to 38.4067680928 at a=2. After some initial terms, the infinite tail of the sum is replaced by the integral of the associated interpolating function, which is available in simple analytic form. Biases that originate from the difference between the smooth area under the function and the corresponding Riemann sum are corrected by standard means. The cases a=3 and a=4 are computed in the same manner.