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Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).
3

%I #30 Feb 24 2022 23:18:53

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

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

%U 3,4,2,7,6,8,0,7,3,5,1,0,0,4,2,2,1,7,5

%N Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).

%C The limiting difference between the integral and sum of x^(1/x). The limit converges slowly.

%H Daniel Hoyt, <a href="/A351885/a351885.png">Graphical representation of the limit.</a>

%H Daniel Hoyt, <a href="/A351885/a351885.pdf">Computing the limiting difference between the sum and integral of x^(1/x).</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes Constants</a>

%F Equals 3/2 - A001620 - A175999 + Sum_{k>=3} Sum_{n>=k} (((-1)^k)*Stieltjes(n)*(n-k+1)^(k-2))/((n-k+2)!*(k-2)!).

%e 0.5681800123590664525123147265218830744...

%o (Python 3)

%o # Gives 15 correct digits

%o from mpmath import stieltjes,fac,quad

%o def limgen(n):

%o terms = []

%o for y in range(3, n):

%o for x in range(y, n):

%o terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))

%o return terms

%o f = lambda x: x**(1/x)

%o int01 = quad(f, [0,1])

%o limit = sum(limgen(60)) + 1.5 - stieltjes(0) - int01

%o print(limit)

%Y Cf. A350862, A329117, A175999, A001620.

%K nonn,cons

%O 0,1

%A _Daniel Hoyt_, Feb 23 2022