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Sophomore's Dream

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Borwein et al. (2004, pp. 4 and 44) term the expression of the integrals

I_1=int_0^1x^xdx
(1)
=0.783430510...
(2)
I_2=int_0^1(dx)/(x^x)
(3)
=1.291285997...
(4)

(OEIS A083648 and A073009) in terms of infinite sums "a sophomore's dream."

For I_1, write

x^x=e^(xlnx)
(5)
=sum_(n=0)^(infty)((xlnx)^n)/(n!)
(6)

Integrating term by term then gives

I_1=sum_(n=0)^(infty)int_0^1((xlnx)^n)/(n!)dx
(7)
=sum_(n=0)^(infty)(-1)^n(n+1)^(-(n+1))
(8)
=sum_(n=1)^(infty)((-1)^(n+1))/(n^n)
(9)

(Borwein et al. 2004, p. 44).

For I_2, write

x^(-x)=e^(-xlnx)
(10)
=sum_(n=0)^(infty)((-xlnx)^n)/(n!)
(11)

Integrating term by term then gives

I_2=sum_(n=0)^(infty)int_0^1((-xlnx)^n)/(n!)dx
(12)
=sum_(n=0)^(infty)(n+1)^(-(n+1))
(13)
=sum_(n=1)^(infty)1/(n^n)
(14)

(Borwein et al. 2004, pp. 4 and 44).

See also

Power Tower

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References

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Dunham, W. The Calculus Gallery: Masterpieces from Newton to Lebesgue. Princeton, NJ: Princeton University Press, pp. 46-51, 2005.Sloane, N. J. A. Sequences A083648 and A073009 in "The On-Line Encyclopedia of Integer Sequences."

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Sophomore's Dream

Cite this as:

Weisstein, Eric W. "Sophomore's Dream." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SophomoresDream.html

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