A342033 Decimal expansion of m(10) = Sum_{n>=0} 1/n!10, the 10th reciprocal multifactorial constant.

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

Offset: 1

Author

Bhoris Dhanjal, Feb 26 2021

Keywords

• nonn
• cons

Comments

m(k) can be proved to approach a harmonic series (and diverge) as k approaches infinity.

Examples

			4.165243765558384590787262...
For n=10, the series is equal to 1+summation from n=1 to 10 (1/n)=9901/2520.
		

Links

• Eric Weisstein's World of Mathematics, Reciprocal Multifactorial Constant

Crossrefs

Cf. A143280 (m(2)), A288055 (m(3)), A288091 (m(4)), A288092 (m(5)), A288093 (m(6)), A288094 (m(7)), A288095 (m(8)), A288096 (m(9)).

Programs

Mathematica

Multifactorial[n_, k_] := Abs[Apply[Times, Range[-n, -1, k]]]
N[Sum[1/Multifactorial[n, 10], {n, 0, 10000}], 105]
(* or *)
ReciprocalFactorialSumConstant[k_] :=
1/k Exp[1/k] (k + Sum[k^(j/k) Gamma[j/k, 0, 1/k], {j, k - 1}])
N[ReciprocalFactorialSumConstant[10], 105]

Formula

m(k) = (1/k)*exp(1/k)*(k + Sum_{j=1..k-1} (k^(j/k)*Gamma(j/k, 1/k))) where Gamma(a,x) the incomplete Gamma function.