A098686 Decimal expansion of Sum_{n >= 1} n/(n^n).

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

Offset: 1

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Oct 27 2004

Keywords

• cons
• nonn

Comments

From Peter Bala, Oct 17 2019: (Start)

Equals 1 + Integral_{x = 0..1} x/x^x dx. More generally, for k = 0,1,2,..., Sum_{n >= k+1} n^k/n^n = Integral_{x = 0..1} x^k/x^x dx.

Also equals the double integral Integral_{x = 0..1, y = 0..1} (1 + x*y)/ (x*y)^(x*y) dx dy. Cf. A073009. (End)

Equals Integral_{x = 0..1} (1 - x*log(x))/x^x dx. - Peter Bala, Jul 21 2022

From Peter Bala, Nov 02 2022: (Start)

Equals Integral_{x = 0..1} (1 + x*log(x)^2)/x^x dx.

Equals the double integral Integral_{x = 0..1, y = 0..1} (x*y*log(x*y) - 1)/( (x*y)^(x*y) * log(x*y) ) dx dy and also equals 1 - Integral_{x = 0..1, y = 0..1} x*y/( (x*y)^(x*y) * log(x*y) ) dx dy by Glasser, Theorem 1. (End)

Examples

			1.62847371290158444705588914326188303165054031095462141647413643009...
		

Links

• M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Crossrefs

Cf. A001113, A013661, A073009.

Programs

Maple

evalf(add(n/(n^n), n = 0..65), 100); # Peter Bala, Nov 02 2022

Mathematica

s = 0; Do[s = N[s + n/n^n, 128], {n, 62}]; RealDigits[s, 10, 111][[1]] (* Robert G. Wilson v, Nov 03 2004 *)

PARI

suminf(n=1, 1/n^(n-1)) \\ Michel Marcus, Oct 21 2019

Extensions

More terms from Robert G. Wilson v, Nov 03 2004