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 A257812 Decimal expansion of Sum_{n=2..infinity} (-1)^n/(n*log(n)). 10
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This alternating series converges quite slowly. However, it can be efficiently computed via its integral representation (see formula below), which converges exponentially fast. This formula and PARI was used to compute 1000 digits. LINKS Iaroslav V. Blagouchine, Table of n, a(n) for n = 0..1000 FORMULA Equals 1/(4*log(2))+2*integral_{x=0..infinity} ((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)). EXAMPLE 0.5264122465333104109306965014111314137217905978875585... MAPLE evalf(sum((-1)^n/(n*log(n)), n=2..infinity), 120); 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); MATHEMATICA NSum[(-1)^n/(n*Log[n]), {n, 2, Infinity}, AccuracyGoal -> 120, WorkingPrecision -> 200, Method -> AlternatingSigns] 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] PROG (PARI) default(realprecision, 120); sumalt(n=2, (-1)^n/(n*log(n))) \\ Vaclav Kotesovec, May 10 2015 (PARI) allocatemem(50000000); 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))) CROSSREFS Cf. A099769, A115563, A257898, A257960, A257964, A257972, A257837. Sequence in context: A308081 A164103 A054863 * A232356 A211015 A077141 Adjacent sequences:  A257809 A257810 A257811 * A257813 A257814 A257815 KEYWORD nonn,cons AUTHOR Iaroslav V. Blagouchine, May 10 2015 STATUS approved

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Last modified June 17 16:48 EDT 2021. Contains 345085 sequences. (Running on oeis4.)