A099769 -id:A099769 - cp's OEIS frontend

A351687 Decimal expansion of Sum_{n>=2} (-1)^n/log(n!).

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

Offset: 1

Bernard Schott, May 05 2022

This series is convergent according to the alternating series test, while series Sum_{n>=2} 1/log(n!) -> infinity (link).

			1.0769010278586314719973748207332879382948126467776416116987...

Socratic Q&A, How do you test the Series Sum_{n>=2} 1/log(n!) for convergence?

Cf. A099769 (Sum_{n>=2} (-1)^n/log(n)).

Cf. A099871, A257898, A257960, A336741.

evalf(sum((-1)^n / log(n!),n=2..infinity),120);

RealDigits[NSum[(-1)^k/Log[k!], {k, 2, Infinity}, WorkingPrecision -> 120, Method -> "AlternatingSigns"]][[1]] (* Amiram Eldar, May 05 2022 *)

sumalt(k=2, (-1)^k/log(k!)) \\ Vaclav Kotesovec, May 05 2022

Equals Sum_{k>=2} (-1)^k/log(k!).

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

Original entry on oeis.org

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

Bernard Schott, Mar 17 2022

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.

			0.8286471276718785080389169468558478918...

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

Cf. A099769, A115563, A168218, A257812.

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

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

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

cp's OEIS Frontend

A351687 Decimal expansion of Sum_{n>=2} (-1)^n/log(n!).

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