Putnam 1997, Problem A-3

Evaluate
$$\int_0^\infty \left( x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \right) \left( 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \right) dx.$$

Solution:
Note that the series on the left is simply \( x \exp(-x^2/2) \). By integration by parts,  
\[
\int_0^\infty x^{2n+1} e^{-x^2/2} dx = 2n \int_0^\infty x^{2n-1} e^{-x^2/2} dx
\]  
and so by induction,  
\[
\int_0^\infty x^{2n+1} e^{-x^2/2} dx = 2 \times 4 \times \cdots \times 2n.
\]  
Thus the desired integral is simply  
\[
\sum_{n=0}^\infty \frac{1}{2^n n!} = \sqrt{e}.
\]