An asymptotic expansion inspired by Ramanujan
Numerical Analysis
2010-05-03 v1 Classical Analysis and ODEs
History and Overview
Abstract
Corollary 2, Entry 9, Chapter 4 of Ramanujan's first notebook claims that a certain sum is asymptotic to ln(x) + gamma, where x is a real variable in the sum and gamma is Euler's constant. Ramanujan's claim is known to be correct for the case n = 1, but incorrect for n > 2 (here n is an integer parameter in the sum). We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a different, correct generalisation of the case n = 1.
Cite
@article{arxiv.1004.5506,
title = {An asymptotic expansion inspired by Ramanujan},
author = {Richard P. Brent},
journal= {arXiv preprint arXiv:1004.5506},
year = {2010}
}
Comments
6 pages. An old Technical Report, submitted for archival purposes. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub139.html