Abstract: Ramanujan suggested an expansion for the n-th partial sum of the harmonic series which employs the reciprocal of the n-th triangular number. The Ramanujan's formula has been proved in 2006 by Villarino, who speculated that there might also exist a similar expansion for the logarithm of the factorial. This study shows that such an asymptotic expansion really exists and provides expressins for its generic coefficient and for the bounds on its errors.
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Some cited references with links:
- - Berndt B.C., Ramanujan's Notebooks: Part V,
Springer Verlag 1998.
- - Copson E.T., Asymptotic Expansions, in Cambridge Tracts in Mathematics, Vol.55,
Paperback Edition, Cambridge University Press 2004. 1st Edition 1965.
- - Abramowitz M., Stegun I.A., Editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,
2nd Edition, Dover Publications 1972,
- - Villarino M.B., Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number,