Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
Abstract
Description
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.
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
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