A336284 -id:A336284 - cp's OEIS frontend

A336730 Decimal expansion of Sum_{n>=1} log(n)^n / n!.

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

Offset: 0

Bernard Schott, Aug 02 2020

With u(n) = log(n)^n / n!, this series is convergent as u(n+1)/u(n) -> 0 when n -> oo.

			0.785672099547734935860778589192856069327...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, 1997, Exercice 3.2.1.r page 279.

Cf. A099871, A306243, A308915, A336284.

evalf(sum(log(n)^n/n!,n=2..infinity),120);

suminf(n=1, log(n)^n/n!) \\ Michel Marcus, Aug 02 2020

Equals Sum_{n>=1} log(n)^n / n!.

A336987 Decimal expansion of Sum_{n>=2} sqrt(n)^log(n)/log(n)^sqrt(n).

Original entry on oeis.org

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

Offset: 2

Bernard Schott, Aug 10 2020

The series u(n) = sqrt(n)^log(n)/log(n)^sqrt(n) is convergent because n^2 * u(n) -> 0 when n -> oo.

			32.219419584243365152435936117722884...

J. Moisan & A. Vernotte, Analyse, Topologie et Séries, Exercices corrigés de Mathématiques Spéciales, Ellipses, 1991, Exercice B-1 a-3 pp. 70, 87-88.

Cf. A099870, A308915, A336284, A336741.

evalf(sum(sqrt(n)^log(n)/log(n)^sqrt(n), n=2..infinity), 120);

default(realprecision, 100); sumpos(n=2, sqrt(n)^log(n)/log(n)^sqrt(n)) \\ Michel Marcus, Aug 10 2020

Equals Sum_{n>=2} sqrt(n)^log(n)/log(n)^sqrt(n).

a(37)-a(101) from Robert Price, Aug 21 2020

cp's OEIS Frontend

A336730 Decimal expansion of Sum_{n>=1} log(n)^n / n!.

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