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

%I #27 Aug 12 2022 19:02:32

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

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

%C 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).

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

%C The sequence converges slowly.

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

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/828323/convergence-of-sum-k-1-infty-sin-left-sqrtk-right-k">Convergence of Sum_{k=0..infinity} sin(sqrt(k)) / k</a>.

%e 1.715671794709...

%o (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

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

%K nonn,cons,more

%O 1,2

%A _Bernard Schott_, May 20 2022

%E More digits from _Stefano Spezia_, May 21 2022