A229020 - OEIS

%I #34 Jun 12 2021 06:10:56

%S 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,

%T 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,

%U 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

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

%C From _Peter Bala_, Jan 28 2015: (Start)

%C 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.

%C 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)

%H G. C. Greubel, <a href="/A229020/b229020.txt">Table of n, a(n) for n = 0..2500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.

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

%e 0.68894844769873820405495001581186710536...

%t 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 *)

%t RealDigits[BesselI[2, 2], 10, 113][[1]] (* _Jean-François Alcover_, Nov 19 2015, after _Peter Bala_ *)

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

%o (PARI) suminf(k=1, k/(k!*(k+1)!)) \\ _Michel Marcus_, Feb 03 2015

%o (PARI) besseli(2, 2) \\ _Altug Alkan_, Nov 19 2015

%Y Cf. A130820.

%K nonn,cons

%O 0,1

%A _Ralf Stephan_, Sep 11 2013