Let f(x) = (x - x^{3}/2 + x^{5}/(2.4) - x^{7}/(2.4.6) + ... ), and g(x) = (1 + x^{2}/2^{2} + x^{4}/(2^{2}4^{2}) + x^{6}/(2^{2}4^{2}6^{2}) + ... ). Find ∫_{0}^{∞} f(x) g(x) dx.

**Solution**

Answer: √e.

*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 x^{2n+1}e^{-x2/2}. This is easily seen to be:

-n! e^{-x2/2}(x^{2n}/n! + 2 x^{2n-2}/(n-1)! + 2^{2}x^{2n-4}/(n-2)! + ... + 2^{n}).

Integrating, only the last term contributes, and we get n! 2^{n}. Thus the x^{2n+1} term integrates to give n! 2^{n}/(n! n! 2^{2n}) = 1/(n! 2^{n}). Hence the answer to the question is the sum from zero to infinity of 1/(n! 2^{n}) which is √e.

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998