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%I #17 Sep 04 2020 03:49:59
%S 7,7,9,5,4,3,3,3,6,0,0,1,6,8,7,7,3,5,0,3,2,9,8,4,5,5,0,2,4,2,0,4,1,9,
%T 0,8,0,1,4,8,8,4,6,3,6,1,5,9,2,1,0,6,0,1,1,9,2,9,5,6,0,5,0,7,4,0,1,4,
%U 5,7,8,0,3,6,0,6,7,8,8,0,4,6,2,4,0,6,0,9,6,7,6,3,0,5,0,7,6,1,2,3,3,3,1,2,3,7,5
%N Decimal expansion of lim_{n->infinity} (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/(2n)! - 1/(2n+1)!))).
%C The sequence of real values x(n) = (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/n! - 1/(n+1)!))) converges to two different limits depending on whether n is even or odd. This integer sequence gives the decimal expansion of the upper limit, to which the even-indexed terms of {x(n)} converge.
%H Rafik Zeraoulia, <a href="https://math.stackexchange.com/q/3698990/156150">Does this a_n = ... have a finite limit?</a>, Math Stackexchange
%e 0.77954333600168773503298455024204190801488463615921...
%t (* note that FullSimplify[1/Factorial[i]-1/Factorial[i+1]] == i/Gamma[2 + i]
%t which is i/Factorial[1 + i] for integer i *)
%t sequence = Table[Fold[#2^#1 &, Table[i/(i + 1)!, {i, n, 1, -1}]], {n, 1, 15}];
%t ListLinePlot[N /@ sequence, PlotRange -> {0, 1}]
%t N[sequence[[-1]]]
%t N[sequence[[-2]]]
%o (PARI) my(N=100, y=(N/(N+1)!)); forstep(n=N-1, 1, -1, y = ((n/(n+1)!)^y)); y \\ _Michel Marcus_, Jul 05 2020
%Y Cf. A328942.
%K nonn,cons
%O 0,1
%A _R Zeraoulia_, Jun 26 2020
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