Ramanujan and Euler's constant
139. R. P. Brent, Ramanujan and Euler's constant,
Proceedings of Symposia in Applied Mathematics, Vol. 48
(edited by W. Gautschi), American Mathematical Society, Providence,
Rhode Island, 1994, 541-545.
MR 95k:01022, MR~95j:00014.
Presented at the American Mathematical Society
"Mathematics of Computation 1943-1993"
Meeting, Vancouver, August 1993.
A longer version appeared as Technical Report CMA-MR02-93/SMS-10-93
"An asymptotic expansion inspired by Ramanujan",
Australian Mathematical Society Gazette 20 (1993), 149-155.
Corollary 2, Entry 9, Chapter 4 of
claims that a certain sum (see the
pdf version of the abstract)
is asymptotic to ln x + gamma
as x -> infinity,
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.
The generalisation mentioned in the abstract was
suggested by Brent and McMillan
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