A394962 Decimal expansion of Sum_{k>=1} Pi/(sqrt(k)*sinh(Pi*sqrt(k))).

3, 5, 0, 8, 5, 2, 8, 4, 4, 9, 6, 5, 1, 3, 5, 3, 6, 0, 1, 6, 8, 5, 9, 3, 4, 4, 6, 7, 9, 7, 9, 5, 8, 2, 3, 8, 8, 7, 9, 4, 2, 2, 3, 9, 5, 1, 4, 4, 9, 0, 1, 3, 3, 8, 6, 2, 8, 5, 8, 1, 9, 9, 3, 9, 9, 6, 9, 7, 2, 3, 5, 5, 5, 0, 9, 0, 7, 1, 7, 8, 4, 7, 3, 2, 1, 9, 5, 4, 3, 7, 4, 4, 3, 4, 6, 0, 6, 7, 2, 7, 5, 7, 2, 5, 1

Offset

0,1

Links

Table of n, a(n) for n=0..104.

H. F. Sandham, Problem 4384, The American Mathematical Monthly, Vol. 57, No. 2 (1950), p. 120; Modified Harmonic Series, Solution to Problem 4384 by the proposer, ibid., Vol. 58, No. 8 (1951), p. 573.

Formula

Equals 1 - 1/2 - 1/3 - 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 - 1/10 - ..., with a change of sign after the reciprocal of each square (Sandham, 1950).

Equals 1 - Sum_{k>=1} (-1)^(k+1) * (H(k+1) - H(k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Example

0.350852844965135360168593448289293938717827305807594...

Mathematica

RealDigits[Pi * NSum[1 / (Sqrt[k] * Sinh[Pi*Sqrt[k]]), {k, 1, Infinity}, WorkingPrecision -> 120]][[1]]

Prog

(PARI) Pi * sumpos(k = 1, 1 / (sqrt(k) * sinh(Pi*sqrt(k))))

Crossrefs

Cf. A001008, A002805.

Cf. A002162, A016655, A262023, A353874.

Sequence in context: A346234 A346479 A346254 * A021289 A200480 A099895

Adjacent sequences: A394959 A394960 A394961 * A394963 A394964 A394965

Keyword

nonn,cons

Author

Amiram Eldar, Apr 08 2026

Status

approved