On the triplicity among infinite products, infinite series, and continued fractions; and its applications to divergent series

TL;DR

This paper explores the concept of triplicity among infinite products, series, and continued fractions, demonstrating how these identities can be used to evaluate both convergent and divergent series with numerous examples.

Contribution

It introduces the concept of triplicity, establishing identities linking infinite products, series, and continued fractions, and applies these to compute values of divergent series.

Findings

Derived identities connecting P, S, and C.

Computed the value of a divergent series from Gauss's diary.

Showcased applications to evaluate divergent series.

Abstract

Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is applied even to obtain sums of divergent series. Many examples of such infinite series are shown, including $1-2+2^3-2^6+\cdots$, which is in Entry 7 of Gauss's diary and its value $0.4275251302\cdots$ is here obtained.