A351738 Decimal expansion of Sum_{k>0} sin(sqrt(k)) / k.

1, 7, 1, 5, 6, 7, 1, 7, 9, 4, 7, 0, 9

Offset: 1

Author

Bernard Schott, May 20 2022

Keywords

• nonn
• cons
• more

Comments

Sum_{k>0} sin(k^alpha) / (k^beta) with 0 < alpha < 1 is convergent if beta > max(alpha, 1-alpha); the constant of this sequence corresponds to the case alpha = 1/2 and beta = 1 (see Arnaudiès).

Consequence: Sum_{k>0} sin(k^(1/m)) / k converges for any positive integer m.

The sequence converges slowly.

Examples

			1.715671794709...
		

References

• J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, 1993, Exercice 11, pp. 316-319.

Links

• Mathematics Stack Exchange, Convergence of Sum_{k=0..infinity} sin(sqrt(k)) / k.

Crossrefs

Cf. A096418, A096444, A114940, A228639, A263193, A342680.

Programs

PARI

default(realprecision, 100); sumalt(k=0, sum(j=1+floor(k^2*Pi^2),floor((k+1)^2*Pi^2), sin(sqrt(j))/j)) \\ Vaclav Kotesovec, May 21 2022

Extensions

More digits from Stefano Spezia, May 21 2022