On Some Continued Fractions and Divergent Series Arising From Integral Families

TL;DR

This paper introduces a method to derive Eulerian continued fractions from integral sequences, reproduces classical expansions, discovers new identities, and proposes a novel summation method for divergent series related to the Euler Mascheroni constant.

Contribution

It presents a new approach to derive continued fractions from integrals and introduces a novel summation method for divergent series, including a new perspective on the Euler Mascheroni constant.

Findings

Reproduces classical continued fractions for logarithm, zeta, and polylogarithms.

Derives new identities for continued fractions.

Proposes a new summation method for divergent series.

Abstract

In this paper we present a method to derive Eulerian continued fractions arising from a sequence of integrals. As examples, through a new derivation, we reproduce classical continued fraction expansions for the natural logarithm, the Riemann zeta function $\zeta(s)$, and polylogarithms, while also obtaining several new identities. Finally, we apply the method to construct a divergent continued fraction, which provides a natural assignment of the Euler Mascheroni constant $\gamma$ as the sum of a particular divergent series through a new summation method which we propose.