Abstract
In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant.
Highlights
College of Engineering and Science, Victoria University, P
We developed many new Euler type identities
We have developed some new identities for the Catalan constant, Apery’s constant and Euler’s famous ζ (2) constant
Summary
We begin by touching on the historical background of Euler sums. The 20th century British mathematician G. In his letter (24th December), he modified his claims stating that they arose out of an error which led to serendipitous discovery of Euler sums: when I recently reconsidered the supposed sums of the two series mentioned at the end of my last letter, I perceived at once that they had arisen by a mere writing mistake. Of this the proverb says “If he had not erred, he should have achieved less”; for on that occasion I came upon the summations of some other series which otherwise I should hardly have looked for, much less discovered.
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