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Decimal expansion of lim_{n->infinity} (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/(2n-1)! - 1/(2n)!))).
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%I #19 Sep 04 2020 03:50:06

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

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

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

%N Decimal expansion of lim_{n->infinity} (1 - 1/2!)^((1/2! - 1/3!)^(...^(1/(2n-1)! - 1/(2n)!))).

%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 lower limit, to which the odd-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 0.54877354704085687513069922740691455562600046738030...

%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=99, y=(N/(N+1)!)); forstep(n=N-1, 1, -1, y = ((n/(n+1)!)^y)); y \\ _Michel Marcus_, Jul 05 2020

%Y Cf. A328941.

%K nonn,cons

%O 0,1

%A _R Zeraoulia_, Jun 26 2020