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A083648
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Decimal expansion of Sum_{n>=1} -(-1)^n/n^n = Integral_{x=0..1} x^x dx.
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24
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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
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OFFSET
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0,1
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COMMENTS
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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.
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REFERENCES
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William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton, NJ 2005, pp. 46-51.
Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton, New Jersey: Princeton University Press (1988), p. 146.
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LINKS
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FORMULA
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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
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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EXAMPLE
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0.78343051071213440705926438652697546940768199014693095825541782270...
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MATHEMATICA
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RealDigits[ Sum[ -(-1)^n /n^n, {n, 1, 60}], 10, 111] [[1]] (* Robert G. Wilson v, Jan 31 2005 *)
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PROG
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(Sage) numerical_approx(-sum((-1/n)^n for n in (1..120)), digits=130) # G. C. Greubel, Mar 01 2019
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CROSSREFS
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Cf. A137420 (continued fraction expansion).
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KEYWORD
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AUTHOR
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STATUS
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approved
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