A354450 -id:A354450 - cp's OEIS frontend

A354592 Decimal expansion of Sum_{k>=1} (1/k - (1 - log(k)/k)^k).

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

Offset: 1

Vaclav Kotesovec, Jun 01 2022

			1.030542353784941208996280929828874607828110554145356713631921644616675...

Cf. A354450, A354593.

Digits := 120: ser := sort(convert(series((1/n - (1 - log(n)/n)^n), n = infinity, 300), polynom), n): s := evalf(sum(op(1, ser), n = 1..infinity), 120): for k from 2 to nops(ser) do serx := expand(op(k, ser)): for j to nops(serx) do s := s + evalf(sum(op(j, serx), n = 1..infinity), 120) end do: print(k, s) end do:

NSum[1/k - (1 - Log[k]/k)^k, {k, 1, Infinity}, WorkingPrecision -> 30, NSumTerms -> 100] (* only 20 digits are correct *)

A354593 Decimal expansion of Sum_{k>=1} (1 - log(k)/k)^(3*k).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 01 2022

			1.109812351672740902597723005686164779380163256103342386792081348419831...

Cf. A354450, A354592.

Digits := 120: ser := sort(convert(series((1-log(n)/n)^(3*n), n = infinity, 300), polynom), n): s := evalf(sum(op(1, ser), n = 1..infinity) + sum(op(2, ser), n = 1..infinity), 120): for k from 3 to nops(ser) do serx := expand(op(k, ser)): for j to nops(serx) do s := s + evalf(sum(op(j, serx), n = 1..infinity), 120) end do: print(k, s) end do:

NSum[(1 - Log[k]/k)^(3*k), {k, 1, Infinity}, WorkingPrecision -> 40, NSumTerms -> 1000]

sumpos(k=1, (1 - log(k)/k)^(3*k)) \\ Michel Marcus, Jun 01 2022

cp's OEIS Frontend

A354592 Decimal expansion of Sum_{k>=1} (1/k - (1 - log(k)/k)^k).

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