Abstract
Ramanujan suggested an expansion for the nth partial sum of the harmonic series which employs the reciprocal of the nth triangular number. This 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 indeed exists and provides formulas for its generic coefficient and for the bounds on its errors.
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Nemes, G. Asymptotic expansion for log n! in terms of the reciprocal of a triangular number. Acta Math Hung 129, 254–262 (2010). https://doi.org/10.1007/s10474-010-0027-5
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DOI: https://doi.org/10.1007/s10474-010-0027-5