Combining Euler series and series from continuations of the Bernoulli scheme

Abstract

This article describes a new approach to constructing numerical series which converge to an irrational number, a two series combination being proposed. The first series is constructed from the probabilistic interpretation of summing numerical series, which is the authors’ continuation of the Bernoulli scheme. Our ideas are based on the rejection of finiteness and independence of trails. A probabilistic urn model allows us to get a convergent numerical series as well as a series coinciding with the elements of Leibniz harmonic triangle diagonals. The second series used in the considered approach is presented by different Euler series which are extensions of the Basel problem. The combination of two numerical series makes it possible to sum unknown numerical series in closed form. A new reading of the Leibniz harmonic triangle and Euler series variations as well as an approach to finding probabilistic urn models of the summing series are proposed. It is symbolic that the paper considers the ways of connecting the “Basel series” and the series obtained from the continuation of the Bernoulli scheme taking into account that Jacob and Johann Bernoulli worked at the University of Basel. The adoption of the method of combining two numerical series enables to obtain results which have not been previously published in scientific literature.