A229020 Decimal expansion of 1 - 1/(1*2) + 1/(1*2*2) - 1/(1*2*2*3) + ...

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

Offset: 0

Author

Ralf Stephan, Sep 11 2013

Keywords

• nonn
• cons

Comments

From Peter Bala, Jan 28 2015: (Start)

As a sum of positive terms, the constant equals Sum_{k >= 1} k/(k!*(k+1)!). If we set S(n) = Sum_{k >= 0} k^n/(k!*(k+1)!) for n >= 0, so this constant is S(1), then S(n) is an integral linear combination of S(0) and S(1). For example S(7) = 16*S(0) + 11*S(1). Cf. A086880. S(0) is A096789.

The Pierce expansion of this constant begins [1, 3, 14, 15, 26, 40, 43, 71, 83, 8120, ...] giving the alternating series representation for this constant 1 - 1/3 + 1/(3*14) - 1/(3*14*15) + 1/(3*14*15*26) - .... (End)

Examples

			0.68894844769873820405495001581186710536...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Eric Weisstein's World of Mathematics, Pierce Expansion.

Crossrefs

Cf. A130820.

Programs

Mathematica

digits = 113; NSum[(-1)^(n+1)*1/Product[1+Floor[k/2], {k, 1, n}], {n, 1, Infinity}, NSumTerms -> digits, Method -> "AlternatingSigns", WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)
RealDigits[BesselI[2, 2], 10, 113][[1]] (* Jean-François Alcover, Nov 19 2015, after Peter Bala *)

PARI

suminf(n=1,(-1)^(n+1)*1./prod(i=1,n,1+floor(i/2)))

PARI

suminf(k=1, k/(k!*(k+1)!)) \\ Michel Marcus, Feb 03 2015

PARI

besseli(2, 2) \\ Altug Alkan, Nov 19 2015

Formula

Equals exp(-2) * Sum_{k>=0} binomial(2*k,k)/(k+1)!. - Amiram Eldar, Jun 12 2021