A352473 Decimal expansion of Sum_{k>=2} (log(k!) / k!).

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

Offset: 0

Author

Bernard Schott, Mar 17 2022

Keywords

• nonn
• cons

Comments

Sum_{k>=2} (log(k) / k) is divergent but here, this series is convergent. If v(k) = log(k!) / k!, we have 0 <= v(k) <= w(k) = k^2/k! with w(k) that is convergent, hence, this positive series is convergent.

Examples

			0.8286471276718785080389169468558478918...
		

References

• J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.13.b p. 250.

Crossrefs

Cf. A099769, A115563, A168218, A257812.

Programs

Maple

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

PARI

sumpos(k=2, log(k!)/k!) \\ Michel Marcus, Mar 17 2022

Formula

Equals Sum_{k>=2} (log(k!) / k!).