Why does $\int_\pi^\infty \frac{2+\cos x} {x}\, \mathrm dx$ diverge

Question

$$\int_\pi^\infty \frac{2+\cos x} {x}\, \mathrm dx,$$ $$0<\frac{1}{x}<\frac{2+\cos x}{x},$$ for $x\geq \pi$.

Can someone explain to me how to this is proved? And what makes the integral diverge? Thank you.

Answer 1

Note that $$\int_\pi^\infty \frac{2+\cos x}{x}dx\ge \int_\pi^\infty \frac{1}{x}dx=\infty$$