A358204 Decimal expansion of Sum_{n >= 1} (-1)^(n+1)/(2*n)^n.
4, 4, 1, 8, 9, 5, 1, 6, 3, 3, 6, 5, 2, 1, 8, 3, 0, 7, 1, 9, 0, 3, 2, 1, 3, 0, 5, 6, 2, 0, 7, 0, 8, 6, 3, 7, 8, 7, 4, 7, 9, 9, 2, 8, 4, 7, 4, 3, 6, 9, 4, 8, 0, 4, 7, 7, 8, 3, 7, 8, 7, 0, 3, 9, 0, 7, 0, 7, 0, 5, 1, 7, 0, 5, 5, 7, 1, 7, 6, 2, 6, 4, 8, 7, 3, 1, 5, 9, 2, 1, 2, 7, 7, 0, 3, 4, 2, 6, 0, 9
Offset: 0
Examples
0.44189516336521830719032130562070863787479928...
Links
- M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
- Eric Weisstein's World of Mathematics, Sophomore's Dream.
Programs
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Maple
evalf( add( (-1)^(n+1)/(2*n)^n, n = 1..50), 100);
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Mathematica
RealDigits[N[Integrate[x^(x/2), {x, 0, 1}]/2, 120]][[1]] (* Amiram Eldar, Jun 21 2023 *) -
PARI
suminf(n=1, (-1)^(n+1)/(2*n)^n) \\ Michel Marcus, Nov 03 2022
Formula
Equals (1/2)*Integral_{x = 0..1} x^(x/2) dx.
Equals (-1/2)*Integral_{x = 0..1} log(x)*(x^(x/2)) dx.
Equals the double integral (1/2)*Integral_{x = 0..1, y = 0..1} (x*y)^(x*y/2) dx dy (apply Glasser, Theorem 1).
Extensions
a(98)-a(99) corrected by Amiram Eldar, Jun 21 2023