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

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.