Suppose we have a function $f(x)$ that goes to zero faster than $1/x$. That is,
$$ \lim_{x \rightarrow 0} xf(x) = 0. $$
I was wondering, does it then follow that the improper integral
$$ \int_1^{\infty} f(x)\, dx $$
converges, assuming that $f$ is continuous and finite on the interval $[1, \infty)$. If not, can you provide a counterexample?
As noted in the comments, $$ \int_1^\infty \frac{1}{x \ln x} dx = \infty.$$ You might then ask what if $f(x) x \ln x \to 0$, does that mean that $\int_1^\infty f dx$ converges? And the answer is no! In fact, $$ \int_1^\infty \frac{1}{x \ln x \ln \ln x} dx = \infty$$ as well. And generally, $$ \int_1^\infty \frac{1}{x \ln x (\ln \ln x) \cdots (\underbrace{\ln \cdots \ln x}_{\text{n many}})} dx = \infty.$$