I am self studying Tom M Apostol Introduction to Analytic number theory.
In theorem 4.12 Apostol uses that improper integral $\int_x^{\infty} \frac {1} { t (logt) ^2 } \ , dt $ converges, x>2 .
I tried using comparison test by comparing with $t^{3/2} $ and $ t^2 $ but I don't get non-zero finite limit of there ratios as t tends to $\infty $ .
Can someone please tell how to prove this integral to be convergent .
The integrand has a simple antiderivative:
$$\int_x^{\infty} \frac{dt}{t (\log{t})^2} = \left [ -\frac1{\log{t}} \right ]_x^{\infty} = \frac1{\log{x}}$$
Note that the integral converges because $\log{t} \to \infty$ as $t \to \infty$.