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%I A083648 #42 Jul 30 2020 01:27:19
%S A083648 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,
%T A083648 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,
%U A083648 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
%N A083648 Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.
%C A083648 In 1697, Johann Bernoulli explores this curve and finds its minimum and the area under the curve from 0 to 1, all this without the benefit of the exponential function.
%D A083648 William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, pp. 46-51.
%D A083648 Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.
%H A083648 G. C. Greubel, <a href="/A083648/b083648.txt">Table of n, a(n) for n = 0..5000</a>
%H A083648 M. L. Glasser, <a href="https://doi.org/10.1080/00029890.2019.1565856">A note on Beukers's and related integrals</a>, Amer. Math. Monthly 126(4) (2019), 361-363.
%H A083648 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H A083648 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SophomoresDream.html">Sophomore's Dream</a>.
%F A083648 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
%F A083648 From _Petros Hadjicostas_, Jun 29 2020: (Start)
%F A083648 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.)
%F A083648 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.)
%F A083648 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)
%e A083648 0.78343051071213440705926438652697546940768199014693095825541782270...
%t A083648 RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* _Robert G. Wilson v_, Jan 31 2005 *)
%o A083648 (PARI) -sumalt(n=1, (-1/n)^(n)) \\ _Michel Marcus_, Oct 15 2015
%o A083648 (Sage) numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # _G. C. Greubel_, Mar 01 2019
%Y A083648 Cf. A137420 (continued fraction expansion).
%Y A083648 Cf. A073009. The minimum point on the curve x^x is (A068985, A072364).
%K A083648 cons,nonn
%O A083648 0,1
%A A083648 _Eric W. Weisstein_, May 01 2003

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