Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
Classical Analysis and ODEs
2007-07-30 v2 General Mathematics
Abstract
An algebraic transformation of the DeTemple-Wang half-integer approximation to the harmonic series produces the general formula and error estimate for the Ramanujan expansion for the nth harmonic number into negative powers of the nth triangular number. We also discuss the history of the Ramanujan expansion for the nth harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.
Keywords
Cite
@article{arxiv.0707.3950,
title = {Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number},
author = {Mark B. Villarino},
journal= {arXiv preprint arXiv:0707.3950},
year = {2007}
}
Comments
sharp error estimates and general formulas for Ramanujan's harmonic number expansion; correction of typo in the Ramanujan-Lodge lower bound constant; thanks to Jonathan Post and Martin Fuller; fixed typo in the title