A354592 -id:A354592 - cp's OEIS frontend

A354450 Decimal expansion of Sum_{k>=1} (1 - log(k)/k)^(2*k).

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

Offset: 1

Vaclav Kotesovec, May 30 2022

Convergence of this sum is slow, but (1 - log(k)/k)^(2*k) can be expanded as 1/k^2 - log(k)^2/k^3 + (-4*log(k)^3 + 3*log(k)^4)/(6*k^4) + (-3*log(k)^4 + 4*log(k)^5 - log(k)^6)/(6*k^5) + ... and each of these summands can be evaluated exactly because Sum_{k>=1} log(k)^r/k^s is equal to the r-th derivative of zeta(s) * (-1)^r. For example, Sum_{k>=1} log(k)^2/k^3 = zeta''(3).

			1.40710442743517658735368769650782855052127407144777551479405092825455...

Mathematica Stack Exchange, Converges or diverges?, 2021.

Cf. A091812, A354592, A354593.

Digits := 120: ser := sort(convert(series((1-log(n)/n)^(2*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[n]/n)^(2*n), {n, 1, Infinity}, WorkingPrecision -> 100, NSumTerms -> 1000] (* only 74 digits are correct *)

default(realprecision, 120); sumpos(k=1, (1 - log(k)/k)^(2*k))

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

A354450 Decimal expansion of Sum_{k>=1} (1 - log(k)/k)^(2*k).

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