A362753 - OEIS

%I #10 May 02 2023 14:40:56

%S 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,

%T 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,

%U 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

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

%C 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).

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

%H Kara Garrison and Thomas E. Price, <a href="https://pillars.taylor.edu/acms-2009/17/">Approximating Sums of Infinite Series</a>, 17th Biennial ACMS Conference MAY 27-30, 2009, Conference Proceedings (2009), pp. 74-83.

%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1112/plms/s2-41.4.257">Notes on the theory of series (xx): On Lambert series</a>, Proc. London Math. Soc., Vol. s2-41, Issue 1 (1936), pp. 257-270.

%H Math Stackexchange, <a href="https://math.stackexchange.com/questions/2161109/does-the-series-sum-k-1-infty-frac-sin1-kk-converge">Does the series Sum_{k=1..n} sin(1/k)/k converge?</a>, 2017.

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

%e 1.47282823195618529629494738382314582532386592787930...

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

%o (PARI) sumpos(k = 1, sin(1/k)/k)

%Y Cf. A233383, A248945, A248946, A362752.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, May 02 2023