Improper integrals $\int_{a_1}^{\infty}\frac{\cos(t)}{t}dt$

Question

Analyze the convergence of:$$\int_{a_1}^{\infty}\frac{\cos(t)}{t}dt$$

and

$$\int_{a_1}^{\infty}\frac{1+\cos(t)}{t}dt$$

and prove that there exists $a_n$ such that

$$\int_{a_{n-1}}^{a_n}\frac{1+\cos(t)}{t}dt=\frac{1}{n}$$

I tried to integrate by parts, but I don't know how to work with this sequence $a_n$