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A099769 Decimal expansion of Sum_{n >= 2} (-1)^n/log(n). 7
9, 2, 4, 2, 9, 9, 8, 9, 7, 2, 2, 2, 9, 3, 8, 8, 5, 5, 9, 5, 9, 5, 7, 0, 1, 8, 1, 3, 5, 9, 5, 9, 0, 0, 5, 3, 7, 7, 3, 3, 1, 9, 3, 9, 7, 8, 8, 6, 9, 1, 9, 0, 7, 4, 7, 7, 9, 6, 3, 0, 4, 3, 7, 2, 5, 0, 7, 0, 0, 5, 4, 1, 7, 1, 1, 4, 3, 4, 6, 8, 9, 7, 9, 8, 9, 9, 1, 3, 4, 7, 6, 6, 4, 9, 4, 6, 9, 1, 9, 5, 3, 5, 7, 4, 1, 4, 5, 2, 8 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

A slowly converging series. The reference (R. E. Shafer) gives several methods for evaluating the sum.

Mathematica program derived from method #3 in the reference (R. E. Shafer). - Ryan Propper, Sep 25 2006

This alternating slowly convergent series may be also efficiently computed via a rapidly convergent integral (see my formula below). I used this formula and PARI to compute 1000 digits of it. - Iaroslav V. Blagouchine, May 11 2015

LINKS

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

R. E. Shafer (proposer), Problem 89-15 SIAM Rev., 32 (1990), 481-483.

FORMULA

Equals 1/(2*log(2)) + 8*integral_{x=0..infinity} arctan(x)/((log(4+4*x^2)^2+4*arctan(x)^2)*sinh(2*Pi*x)). - Iaroslav V. Blagouchine, May 11 2015

EXAMPLE

0.9242998972229388559595701813595900537733193978869190...

MAPLE

evalf(sum((-1)^n/log(n), n=2..infinity), 120); # Vaclav Kotesovec, May 11 2015

MATHEMATICA

Do[X = 2*i; T = Table[Table[0, {X}], {X}]; For[n = 2, n <= X, n++, T[[n, 2]] = Sum[(-1)^k/Log[k], {k, 2, n}]]; For[k = 2, k <= X, k++, For[n = 2, n <= X - k + 1, n++, T[[n, k+1]] = T[[n+1, k-1]] + 1/(T[[n+1, k]] - T[[n, k]])]]; Print[N[T[[2, X]], 50]], {i, 50}] (* Ryan Propper, Sep 25 2006 *)

digits = 105; NSum[(-1)^n/Log[n], {n, 2, Infinity}, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 12 2013 *)

1/(2*Log[2])+8*NIntegrate[ArcTan[x]/((Log[4+4*x^2]^2+4*ArcTan[x]^2)*Sinh[2*Pi*x]), {x, 0, Infinity}, WorkingPrecision -> 109] // RealDigits // First (* Jean-François Alcover, May 12 2015, after Iaroslav V. Blagouchine *)

PROG

(PARI) sumalt(n=2, (-1)^n/log(n)) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

(PARI) allocatemem(50000000);

  default(realprecision, 1100);  1/(2*log(2)) + intnum(x=0, 1000, 8*atan(x)/((log(4+4*x^2)^2+4*atan(x)^2)*sinh(2*Pi*x))) \\ Iaroslav V. Blagouchine, May 11 2015

CROSSREFS

Cf. A257837, A257964, A257972, A257898, A257960, A257812.

Sequence in context: A248320 A200282 A133841 * A176517 A020784 A111188

Adjacent sequences:  A099766 A099767 A099768 * A099770 A099771 A099772

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Nov 11 2004

EXTENSIONS

a(9)-a(17) from Ryan Propper, Sep 25 2006

a(18)-a(104) from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)