I am working on some exercises for Improper Integrals (not homework). The question is 1.c in Folland Advanced Calculus :
$$\int_0^\infty x^2 e^{-x^2 } \, dx$$
It asks whether the above Improper Integral is convergent. Folland's answer to this is that it is convergent. I however cant see how that could be. If I use the theorem (4.55 in text) that states:
$$0\le f\left( x \right) \le g(x) \text{ for all sufficiently large }x. \\\text{If }\int_0^\infty g(x) \, dx \text{ converges so does }\int_0^\infty f(x) \, dx.$$
then I just can't think of a $g(x)$ that would satisfy this.