1,1

Decimal expansion is A131917. 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 n-th harmonic number into negative powers of the n-th triangular number. We also discuss the history of the Ramanujan expansion for the n-th harmonic number as well as sharp estimates of its accuracy, with complete proofs, and we compare it with other approximative formulas.

Mark B. Villarino, Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number. Constant occurs in Theorem 7 (DeTemple-Wang), formula (1.14), page 6.

(54 ln(3/2) + 54 gamma - 53)/(1 - ln(3/2) - gamma) = 1 / (1 - gamma - ln(3/2)) - 54, where Martin Fuller simplifies the constant which Villarino showed was implicitly given by DeTemple and Wang.

3.73929751945... = 3 + 1/1+ 1/2+ 1/1+ 1/5+ 1/11+ 1/7+ 1/6+ 1/1+ 1/2+ 1/6+ 1/1+ 1/10+ 1/15+ 1/7+ 1/1+ 1/11+ 1/12+ 1/1+ 1/1+ 1/4+ 1/3+ 1/1+ 1/1+ 1/9+ 1/3+ 1/4+ 1/10+ 1/4+ 1/1+ 1/1+ 1/26+ 1/1+ 1/1+ 1/8+ 1/10+ 1/1+ 1/2+ 1/1+ 1/1+ 1/1+ 1/2+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/3+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/1+ 1/3+ 1/1+ 1/1+ 1/1+ 1/2+ 1/4+ 1/2+ 1/1+ 1/49+ 1/7+ 1/1+ 1/2+ 1/1+ 1/1+ 1/2+ 1/16+ 1/1+ 1/283+ 1/1+ 1/1+ 1/5+ 1/1+ 1/1+ 1/1+ 1/2+ 1/1+ 1/30+ 1/19+ 1/1+ 1/11+ 1/2+ 1/5+ 1/10+ 1/3+ 1/1+ 1/4+ 1/1+ 1/6+ 1/2+ 1/19+ 1/1+ 1/1+ 1/3+ 1/2+ 1/1+ . . .

Cf. A001008, A001620, A131915, A131916, A131917.

Adjacent sequences: A131915 A131916 A131917 this_sequence A131919 A131920 A131921

Sequence in context: A059807 A135261 A102774 this_sequence A010123 A039620 A008296

cofr,cons,easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Jul 27 2007