A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.
7, 8, 3, 4, 3, 0, 5, 1, 0, 7, 1, 2, 1, 3, 4, 4, 0, 7, 0, 5, 9, 2, 6, 4, 3, 8, 6, 5, 2, 6, 9, 7, 5, 4, 6, 9, 4, 0, 7, 6, 8, 1, 9, 9, 0, 1, 4, 6, 9, 3, 0, 9, 5, 8, 2, 5, 5, 4, 1, 7, 8, 2, 2, 7, 0, 1, 6, 0, 0, 1, 8, 4, 5, 8, 9, 1, 4, 0, 4, 4, 5, 6, 2, 4, 8, 6, 4, 2, 0, 4, 9, 7, 2, 2, 6, 8, 9, 3, 8, 9, 7, 4, 8, 0, 0
Offset: 0
Examples
0.78343051071213440705926438652697546940768199014693095825541782270...
References
- William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, pp. 46-51.
- Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
- Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- 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, Power Tower.
- Eric Weisstein's World of Mathematics, Sophomore's Dream.
Crossrefs
Programs
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Mathematica
RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* Robert G. Wilson v, Jan 31 2005 *) -
PARI
-sumalt(n=1, (-1/n)^(n)) \\ Michel Marcus, Oct 15 2015
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Sage
numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # G. C. Greubel, Mar 01 2019
Formula
Constant also equals the double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy. - Peter Bala, Mar 04 2012
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals -Integral_{x=0..1, y=0..1} (x*y)^(x*y)/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to Integral_{x=0..1} x^x dx.)
Equals -Integral_{x=0..1} x^x*log(x) dx. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the double integral of Peter Bala above.)
Without using the results in Glasser (2019), notice that Integral x^x*(1 + log(x)) dx = x^x + c, which implies Integral_{x=0..1} x^x dx = -Integral_{x=0..1} x^x*log(x) dx. (End)
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