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A257812 Decimal expansion of Sum_{n=2..infinity} (-1)^n/(n*log(n)). 8
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: A078716 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 August 19 13:25 EDT 2018. Contains 313862 sequences. (Running on oeis4.)