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