Divergent series in Gauss's diary and their extensions

TL;DR

This paper systematically extends two divergent series from Gauss's diary using parameters and applies classical identities to find equivalent series, enabling the computation of their sums.

Contribution

It introduces a systematic extension method for divergent series in Gauss's diary and applies classical identities to evaluate their sums.

Findings

Extended divergent series with additional parameters.

Derived equivalent series using Rogers-Fine identities.

Computed sums of the divergent series.

Abstract

Two divergent series in Entry 7 of Gauss's diary are extended systematically by introducing additional parameters. Rogers-Fine identities, Ramanujan's continued fractions and Heine's transformation relations of basic hypergeometric series are applied to find equivalent alternative series, which enable us to compute the sums of divergent series in question.