Sophomore's Dream

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

			(1)
			(2)
			(3)
			(4)

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

For , write

			(5)
			(6)

Integrating term by term then gives

			(7)
			(8)
			(9)

(Borwein et al. 2004, p. 44).

For , write

			(10)
			(11)

Integrating term by term then gives

			(12)
			(13)
			(14)

(Borwein et al. 2004, pp. 4 and 44).

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."