Some series related to infinite series given by Ramanujan

Abstract

In this paper we discuss some formulas concerning the summation of certain infinite series, given by Ramanujan in his notebooks [1], vol. 1, Ch. XVI (pp. 251–263), and vol. 2, Ch. XV (pp. 181–192). (A large part of the material in Ch. XVI is contained also in Ch. XV, with only minor changes.) It is shown that several of the formulas given are erroneous. Most of the remaining formulas have by now been proved by residuum calculus. Some of these proofs are extendable to cases which do not seem to have attracted attention earlier. As an example of this we mention the sums $$\sum\limits_{n = 1}^\infty {( - 1)^{n - 1} n^s } /\sinh n\pi = 0 for s = 5,9,13,17,...$$