Proof the convergence of this improper integral

Question

I would like to prove $$\int_{1}^{\infty}\frac{\cos x}{x}dx$$ converges by first establishing the convergence of the improper integral of $$\int_{1}^{\infty} \frac{1}{x^{2}}dx$$The latter is easy but what is the connection with the former? Thanks a lot!

Answer 1

Hint: use integration by parts, with $u = \frac{1}{x}$ and $dv = \cos(x)dx$. You will obtain a term whose limit you know how to evaluate, and an integral term which can be bounded by $$\int_{1}^{\infty} \frac{1}{x^{2}}dx$$ If you need further help, feel free to ask.