Asymptotic expansion for log n! in terms of the reciprocal of a triangular number

G. Nemes

doi:10.1007/s10474-010-0027-5

Asymptotic expansion for log n! in terms of the reciprocal of a triangular number

G. Nemes^*

^*Corresponding author for this work

Eötvös Loránd University

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

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.

Original language	English
Pages (from-to)	254-262
Number of pages	9
Journal	Acta Mathematica Hungarica
Volume	129
Issue number	3
DOIs	https://doi.org/10.1007/s10474-010-0027-5
State	Published - Nov 2010
Externally published	Yes

Keywords

Ramanujan's formula
approximation
asymptotic expansion
factorial

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10.1007/s10474-010-0027-5

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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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AB - 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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KW - approximation

KW - asymptotic expansion

KW - factorial

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ER -

Asymptotic expansion for log n! in terms of the reciprocal of a triangular number

Abstract

Keywords

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