A274031 Decimal expansion of a doubly infinite sum involving harmonic numbers. Curiously, this sum is very close to 1.

9, 9, 9, 2, 2, 2, 8, 3, 7, 7, 6, 3, 8, 3, 0, 0, 0, 8, 7, 6, 1, 9, 3, 5, 7, 4, 9, 2, 4, 7, 5, 6, 9, 8, 8, 6, 0, 3, 6, 9, 9, 5, 5, 1, 6, 1, 3, 6, 1, 7, 0, 9, 4, 4, 2, 0, 4, 8, 9, 8, 4, 3, 5, 8, 6, 2, 7, 6, 1, 0, 2, 2, 9, 7, 3, 5, 5, 0, 1, 2, 4, 2, 2, 2, 1, 9, 6, 3, 5, 3, 5, 0, 3, 5, 5, 9, 7, 6, 4, 7, 3, 9, 2

Offset

0,1

Links

Table of n, a(n) for n=0..102.

R. Pemantle, C. Schneider, When is 0.999... equal to 1?, arXiv:math/0511574 [math.CO], 2005.

Formula

Sum_{j >= 1, k >= 1} H(j) (H(k+1)-1)/(j k (k+1) (j+k)), where H(j) is the j-th harmonic number.

Equals -4 zeta(2) - 2 zeta(3) + 4 zeta(2) zeta(3) + 2 zeta(5).

Example

0.99922283776383000876193574924756988603699551613617094420489843586276...

Mathematica

RealDigits[-4 Zeta[2] - 2 Zeta[3] + 4 Zeta[2] Zeta[3] + 2 Zeta[5], 10,
   103][[1]]

Crossrefs

Sequence in context: A346929 A346927 A197149 * A111623 A229985 A019897

Adjacent sequences: A274028 A274029 A274030 * A274032 A274033 A274034

Keyword

nonn,cons

Author

Jean-François Alcover, Jun 07 2016

Status

approved