# Convergence of infinite series of log function [closed]

Check the convergence of the infinite series $$\sum\limits\frac1{(\log n)^{3/2}}$$.

I have tried to use comparison test but got no success.

• Hint:$$(\log{(n)})^{3/2}\lt n\qquad\forall n\in\mathbb{N}$$ Commented Jul 11, 2020 at 14:00
• What have you tried explicitly? Where are you stuck? See How to ask a good question. Commented Jul 11, 2020 at 14:02
• Have you learned the Cauchy Condensation test? That makes the term you're summing over immediately much more manageable. Commented Jul 11, 2020 at 14:14
• Thanks for replying 😊. I used the comparison test to prove that 1/log n is divergent (with 1/n). I wasn't able to get such divergent series to make a conclusion. Commented Jul 11, 2020 at 14:26
• @ Peter Foreman Thank u for the hint. can you also give a proof for this identity? Commented Jul 11, 2020 at 14:29

More generally, For $$\sum\limits_{n=2}^\infty \frac{1}{(\operatorname{log}n)^p}$$
$$\begin{eqnarray} \sum\limits_{n=2}^\infty 2^n \frac{1}{(\operatorname{log}2^n)^p}=\frac{1}{(\operatorname{log}2)^p} \sum\limits_{n=2}^\infty \frac{2^n}{n^p} \end{eqnarray}$$
which diverges for all $$p>0$$. Since $$2^nn^{-p}\to \infty$$ for all $$p>0$$