cp's OEIS Frontend

A257812 Decimal expansion of Sum_{n>=2} (-1)^n/(n*log(n)).

%I A257812 #39 Mar 01 2025 16:45:42
%S A257812 5,2,6,4,1,2,2,4,6,5,3,3,3,1,0,4,1,0,9,3,0,6,9,6,5,0,1,4,1,1,1,3,1,4,
%T A257812 1,3,7,2,1,7,9,0,5,9,7,8,8,7,5,5,8,5,4,0,7,4,6,9,9,5,7,0,0,8,3,3,7,8,
%U A257812 3,2,2,3,1,3,0,2,0,8,4,4,6,9,8,4,6,3,6,2,2,7,2,9,7,3,4,6,1,5,1,7,8,8,7,6,4,9,5,5,8
%N A257812 Decimal expansion of Sum_{n>=2} (-1)^n/(n*log(n)).
%C A257812 This alternating series converges quite slowly. However, it can be efficiently computed via its integral representation (see formula below), which converges exponentially quickly. This formula and PARI were used to compute 1000 digits.
%H A257812 Iaroslav V. Blagouchine, <a href="/A257812/b257812.txt">Table of n, a(n) for n = 0..1000</a>
%F A257812 Equals 1/(4*log(2)) + 2*Integral_{x=0..oo} (2*arctan(x)+x*log(4+4*x^2))/(sinh(2*Pi*x)*(log(4+4*x^2)^2+4*arctan(x)^2)*(x^2+1)) dx.
%e A257812 0.5264122465333104109306965014111314137217905978875585...
%p A257812 evalf(sum((-1)^n/(n*log(n)), n=2..infinity), 120);
%p A257812 evalf(1/(4*log(2))+2*(Int((2*arctan(x)+x*log(4+4*x^2))/(sinh(2*Pi*x)*(log(4+4*x^2)^2+4*arctan(x)^2)*(x^2+1)), x=0..infinity)), 120);
%t A257812 NSum[(-1)^n/(n*Log[n]), {n, 2, Infinity}, AccuracyGoal -> 120, WorkingPrecision -> 200, Method -> AlternatingSigns]
%t A257812 1/(4*Log[2])+2*NIntegrate[(2*ArcTan[x]+x*Log[4+4*x^2])/((x^2+1)*Sinh[2*Pi*x]*(Log[4+4*x^2]^2+4*ArcTan[x]^2)), {x, 0,Infinity}, WorkingPrecision->120]
%o A257812 (PARI) default(realprecision,120); sumalt(n=2, (-1)^n/(n*log(n))) \\ _Vaclav Kotesovec_, May 10 2015
%o A257812 (PARI) allocatemem(50000000);
%o A257812 default(realprecision, 1200); 1/(4*log(2))+2*intnum(x=0, 1000, (2*atan(x)+x*log(4+4*x^2))/(sinh(2*Pi*x)*(log(4+4*x^2)^2+4*atan(x)^2)*(x^2+1)))
%Y A257812 Cf. A099769, A115563, A257898, A257960, A257964, A257972, A257837.
%K A257812 nonn,cons
%O A257812 0,1
%A A257812 _Iaroslav V. Blagouchine_, May 10 2015