Let f(x) = (x - x3/2 + x5/(2.4) - x7/(2.4.6) + ... ), and g(x) = (1 + x2/22 + x4/(2242) + x6/(224262) + ... ). Find ∫0∞ f(x) g(x) dx.
Fairly easy. A routine computation, once you have recognized a closed form for one of the series.
We notice that the first series is x e-x2/2. The second series is clearly absolutely convergent and so we can multiply out and integrate each term separately. Thus we have to integrate x2n+1e-x2/2. This is easily seen to be:
-n! e-x2/2(x2n/n! + 2 x2n-2/(n-1)! + 22x2n-4/(n-2)! + ... + 2n).
Integrating, only the last term contributes, and we get n! 2n. Thus the x2n+1 term integrates to give n! 2n/(n! n! 22n) = 1/(n! 2n). Hence the answer to the question is the sum from zero to infinity of 1/(n! 2n) which is √e.
© John Scholes
12 Dec 1998