Q:

Comparison test on $ \int_2 ^\infty \frac{\ln(t)}{t^{3/2}} \ dt$

How would one use the comparison test to deduce whether the following function converges or diverges?
I’ve attempted to use $ \ln(t) < t $ for $ t > 1$ but fail to conclude whether the desired function converges.
I’ve also attempted to use $ \ln (t) > 1 $ for $ t > e $ but also encounter a similar issue once again. 

A:

The standard trick is use limit comparison test with $\frac{1}{t^{\alpha}} $ for $1<\alpha<\frac32$, as for example by the convergent
$$\int_2 ^\infty \frac{1}{t^{5/4}} \ dt$$
indeed as $x\to \infty$
$$\frac{\frac{\ln t}{t^{3/2}}}{\frac{1}{t^{5/4}} }=\frac{\ln t}{t^{1/4}}\to 0 $$
and thus the given integral converges.

