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Decimal expansion of Sum_{k>=1} 1/Fibonacci(k!).
1

%I #7 Mar 12 2024 02:47:20

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

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

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

%N Decimal expansion of Sum_{k>=1} 1/Fibonacci(k!).

%C Nyblom (2000) proved that this constant is transcendental.

%H M. A. Nyblom, <a href="https://doi.org/10.1216/rmjm/1021477261">A theorem on transcendence of infinite series</a>, The Rocky Mountain Journal of Mathematics, Vol. 30, No. 3 (2000), pp. 1111-1120; <a href="https://www.jstor.org/stable/44238526">alternative link</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals Sum_{k>=1} 1/A063374(k).

%e 2.12502156659765355417529349235237991793625797423002...

%t RealDigits[Sum[1/Fibonacci[k!], {k, 1, 10}], 10, 120][[1]]

%o (PARI) suminf(k = 1, 1/fibonacci(k!))

%Y Cf. A000045, A000142, A063374, A343202, A371137.

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Mar 12 2024