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%I #27 Mar 23 2020 17:32:46
%S 5,4,2,6,6,6,7,3,2,5,7,0,2,8,2,7,5,4,2,8,8,8,5,0,7,4,7,6,3,9,6,2,4,7,
%T 4,8,7,9,1,4,2,0,3,6,3,7,6,3,0,9,2,7,2,0,0,9,5,0,7,8,6,6,0,1,3,8,1,0,
%U 1,1,7,9,9,6,4,3,2,3,3,3,6,7,3,6,3,9,8,3,4,5,7,0,2,2,3,6,5,4,2,0,4,8,2,8,6,3,8,5,5
%N Decimal expansion of Sum_{n=1..infinity} (-1)^(n-1)/(n - log(n)).
%C This alternating series converges quite slowly, but can be efficiently computed via its integral representation (see my formula below), which converges exponentially fast. I used this formula and PARI to compute 1000 digits of this series. Modern CAS are also able to evaluate it very quickly and to a high degree of accuracy.
%H Iaroslav V. Blagouchine, <a href="/A257972/b257972.txt">Table of n, a(n) for n = 0..1000</a>
%F Equals 1/2 + integral_{x=0..infinity} (x-arctan(x))/(sinh(Pi*x)*((1-1/2*log(1+x^2))^2 + (x-arctan(x))^2)).
%e 0.542666732570282754288850747639624748791420363763092...
%p evalf(sum((-1)^(n-1)/(n-log(n)), n = 1..infinity), 120);
%p evalf(1/2+Int((x-arctan(x))/(sinh(Pi*x)*((1-(1/2)*log(1+x^2))^2+(x-arctan(x))^2)), x = 0..infinity), 120);
%t N[NSum[(-1)^(n-1)/(n-Log[n]), {n, 1, Infinity}, AccuracyGoal -> 120, Method -> "AlternatingSigns", WorkingPrecision -> 200],119]
%t N[1/2 + NIntegrate[(x-ArcTan[x])/(Sinh[Pi*x]*((1-1/2*Log[1+x^2])^2 + (x-ArcTan[x])^2)), {x, 0, 1, Infinity}, AccuracyGoal -> 120, WorkingPrecision -> 200],119] (* The integrand reaches a local maximum near x=1.02, so for better numerical accuracy, split the interval of integration into two or three parts. *)
%o (PARI) default(realprecision, 120); sumalt(n=1, (-1)^(n-1)/(n-log(n)))
%o (PARI) allocatemem(50000000);
%o default(realprecision, 1200); 1/2 + intnum(x=0,1, (x-atan(x))/(sinh(Pi*x)*((1-0.5*log(1+x^2))^2 + (x-atan(x))^2))) + intnum(x=1,3, (x-atan(x))/(sinh(Pi*x)*((1-0.5*log(1+x^2))^2 + (x-atan(x))^2))) + intnum(x=3,1000, (x-atan(x))/(sinh(Pi*x)*((1-0.5*log(1+x^2))^2 + (x-atan(x))^2))) /* The integrand reaches a local maximum near x=1.02, so for better numerical accuracy, split the interval of integration into two or three parts. */
%o (Sage)
%o from mpmath import mp, nsum, inf
%o mp.dps = 110; mp.pretty = True
%o nsum(lambda n: (-1)^(n-1)/(n-log(n)), [1, inf], method='alternating') # _Peter Luschny_, May 17 2015
%Y Cf. A099769, A257837, A257812, A257898, A257960, A257964.
%K nonn,cons
%O 0,1
%A _Iaroslav V. Blagouchine_, May 15 2015
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