Chapter 1.16, Problem 3MP

Show that ${\displaystyle {\sum }_{n=2}^{\infty }1/n^{3/2}}$ is convergent. What is wrong with the following "proof" that it diverges? $\frac{1}{\sqrt{8}}+\frac{1}{\sqrt{27}}+\frac{1}{\sqrt{64}}+\frac{1}{\sqrt{125}}+\mathrm{...}>\frac{1}{\sqrt{9}}+\frac{1}{\sqrt{36}}+\frac{1}{\sqrt{81}}+\frac{1}{\sqrt{144}}+\mathrm{...}$

which is

$\frac{1}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\mathrm{...}=\frac{1}{3}\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\mathrm{...}\right)$

Since the harmonic series diverges, the original series diverges. Hint: Compare 3n and $n\sqrt{n}$.