Representations of functions in series with parameter

TL;DR

This paper presents a general theorem that transforms Taylor series into parameter-dependent series, unifying classical results and deriving a new representation for Euler's constant gamma.

Contribution

It introduces a broad theorem that converts Taylor series into parameter-dependent series, encompassing classical results and providing a novel representation for gamma.

Findings

Unified classical results through the theorem

Derived a new series representation for gamma

Transformations applicable to various Taylor series

Abstract

We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for the Euler constant gamma. The theorem transforms every Taylor series into a series depending on a parameter.