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

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

Offset: 0

Author

Iaroslav V. Blagouchine, May 10 2015

Keywords

• nonn
• cons

Comments

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.

Examples

			0.5264122465333104109306965014111314137217905978875585...
		

Links

• Iaroslav V. Blagouchine, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A099769, A115563, A257898, A257960, A257964, A257972, A257837.

Programs

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]

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)))

Formula

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.