I am looking to establish the convergence of $$\sum_{n\geq 1}\frac{\log(n)}{n^{\frac{3}{2}}}.$$ I have seen a previous article (Prove the convergence of : $\sum \ln(n)/n^{3/2}$) looking into this.
The Cauchy Condensation Test is not something I would be able to use yet, as it is not taught.
I see assertions without proof that (*) for all $\alpha \gt 0$, $\log(n)\lt n^\alpha$ eventually, and using the p-test, this solves the problem. How can I prove such a result?
Another method I tried was using the integral test, but here I am assuming that the expression is decreasing as $n$ increases, and I find that the convergence is dependent on whether $$\frac{\log(n)}{n^\frac{1}{2}}$$ converges, and now I have a similar problem to before.
I am looking in particular for a proof of (*), but any alternate methods with proof would be great too! Thank you
