A351885 Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).

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

Offset: 0

Author

Daniel Hoyt, Feb 23 2022

Keywords

• nonn
• cons

Comments

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

Examples

			0.5681800123590664525123147265218830744...
		

Links

• Daniel Hoyt, Graphical representation of the limit.
• Daniel Hoyt, Computing the limiting difference between the sum and integral of x^(1/x).
• Wikipedia, Stieltjes Constants

Crossrefs

Cf. A350862, A329117, A175999, A001620.

Programs

Python

# Gives 15 correct digits
from mpmath import stieltjes,fac,quad
def limgen(n):
    terms = []
    for y in range(3, n):
        for x in range(y, n):
            terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))
    return terms
f = lambda x: x**(1/x)
int01 = quad(f, [0,1])
limit = sum(limgen(60)) + 1.5 - stieltjes(0) - int01
print(limit)

Formula

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)!).