We must be very careful with what we mean by "converges" when talking about improper integrals. The usual definition of $\int_a^b f(x)dx$ converging is that $\int_a^b f^+(x) dx$ and $\int_a^b f^-(x)dx$ are both finite, where $f^+(x) = \max\{f(x),0\}$ and $f^-(x) = \max\{-f(x),0\}$.
In the case of $\mathbb{R}$ (as opposed to $\mathbb{R}^n$) if $a$ or $b$ is infinite it is common to say an integral converges if, e.g. the case where $b$ is infinite $\lim_{M \to \infty} \int_a^M f(x)dx$ exists in $\mathbb{R}$.
For the case you present we can talk about another kind of convergence, namely existence of the principal value of the integral. (Most would not call this "convergence" though I could see why some might.)
In your case, I would use the first (i.e. the most standard) definition of convergence to conclude that the integral diverges.