A307671 - OEIS

%I #41 May 02 2019 06:09:44

%S 2,7,2,3,4,3,5,8,7,7,0,7,5,9,6,7,6,4,7,8,4,0,7,0,6,7,6,9,2,3,9,5,5,5,

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

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

%N Decimal expansion of the alternating convergent series S = Sum_{k>=0} (-1)^k*f(k), where f(k) = harmonic(2^k) - k*log(2) - gamma, harmonic(m) is the Sum_{j=1..m} 1/j, and gamma is Euler-Mascheroni constant.

%e 0.272343587707596764784070676923955578748225108064395871645389620412837597...

%p evalf(Sum((-1)^k*(harmonic(2^k) - k*log(2) - gamma), k=0..infinity), 120); # _Vaclav Kotesovec_, Apr 30 2019

%t digits = 104; s = NSum[(-1)^k*(HarmonicNumber[2^k] - k*Log[2] - EulerGamma), {k, 0, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]; RealDigits[s, 10, digits][[1]] (* _Jean-François Alcover_, Apr 28 2019 *)

%o (PARI) default(realprecision, 120); sumalt(k=0, (-1)^k*(psi(2^k+1) - k*log(2))) \\ _Vaclav Kotesovec_, Apr 30 2019

%Y Cf. A001620 (Euler-Mascheroni), A001008/A002805 (harmonic), A002162 (log(2)), A094640 (alternate Euler's constant), A256921 (a similar constant).

%K cons,nonn

%O 0,1

%A _Luis H. Gallardo_, Apr 20 2019