On Some Continued Fractions and Divergent Series Arising From Integral Families

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.