A362753 Decimal expansion of Sum_{k>=1} sin(1/k)/k.

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

Offset: 1

Author

Amiram Eldar, May 02 2023

Keywords

• nonn
• cons

Comments

The value of the Hardy-Littlewood function H(x) = Sum_{k>=1} sin(x/k)/k at x = 1 (Hardy and Littlewood, 1936; Gautschi, 2004).

Examples

			1.47282823195618529629494738382314582532386592787930...
		

References

• Walter Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, 2004. See Example 3.64, pp. 242-245.

Links

• Kara Garrison and Thomas E. Price, Approximating Sums of Infinite Series, 17th Biennial ACMS Conference MAY 27-30, 2009, Conference Proceedings (2009), pp. 74-83.
• G. H. Hardy and J. E. Littlewood, Notes on the theory of series (xx): On Lambert series, Proc. London Math. Soc., Vol. s2-41, Issue 1 (1936), pp. 257-270.
• Math Stackexchange, Does the series Sum_{k=1..n} sin(1/k)/k converge?, 2017.

Crossrefs

Cf. A233383, A248945, A248946, A362752.

Programs

Maple

evalf(sum(sin(1/k)/k, k = 1 .. infinity), 120);

PARI

sumpos(k = 1, sin(1/k)/k)

Formula

Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k)/(2*k-1)!.