An asymptotic expansion inspired by Ramanujan

Details An asymptotic expansion inspired by Ramanujan An asymptotic expansion inspired by Ramanujan

2010-04-30

arxiv.org/abs/1004.5506v1

Australian Mathematical Society Gazette 20 (1993), 149-155.

Mathematics

Numerical Analysis

6 pages. An old Technical Report, submitted for archival purposes. For further details, see http://wwwmaths.anu.edu.au/~brent/

Scientific paper

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.

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