Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series

Details Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series Ramanujan's Approximation to the nth Partial Sum of the Harmonic Series

2004-02-22

arxiv.org/abs/math/0402354v5

Mathematics

Classical Analysis and ODEs

7 pages. Numerical copy error corrected as well as the computations based on it. Minor expository improvements

Scientific paper

A simple integration by parts and telescopic cancellation leads to a rigorous
derivation of the first 2 terms for the error in Ramanujan's asymptotic series
for the nth partial sum of the harmonic series. Then Kummer's transformation
gives three more terms and a rigorous error estimate. Finally best-possible
estimates of Lodge's approximations.

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