Does $\infty$ mean $+\infty$ in "English mathematics"?

In the context of real numbers it is a relatively safe assumption that an author writing that a sequence $(x_n)$ tends to $\infty$ means that for each $M>0$ there is an $n_0$ such that $x_n \ge M$ for $n \ge n_0$, and not $|x_n| \ge M$ for $n \ge n_0$. If the latter is meant it would usually be indicated.

In the context of complex numbers it will of course be different.