Answer
$\mathop \sum \limits_{n = 10}^\infty \dfrac{1}{{n{{\left( {\ln n} \right)}^{3/2}}}}$ converges.
Work Step by Step
Write $f\left( x \right) = \dfrac{1}{{x{{\left( {\ln x} \right)}^{3/2}}}}$
For $x \ge 2$, $f$ is a positive, decreasing, and continuous function of $x$ (please see the figure attached), so we can use the Integral Test.
Write $t = \ln x$. So, $dt = \dfrac{1}{x}dx$.
$\mathop \smallint \limits_2^\infty \dfrac{1}{{x{{\left( {\ln x} \right)}^{3/2}}}}{\rm{d}}x = \mathop \smallint \limits_{\ln 2}^\infty \dfrac{1}{{{t^{3/2}}}}{\rm{d}}t = - \dfrac{2}{{\sqrt t }}|_{\ln 2}^\infty = \dfrac{2}{{\sqrt {\ln 2} }}$
Since $\mathop \smallint \limits_2^\infty \dfrac{1}{{x{{\left( {\ln x} \right)}^{3/2}}}}{\rm{d}}x$ converges, by the Integral Test $\mathop \sum \limits_{n = 10}^\infty \dfrac{1}{{n{{\left( {\ln n} \right)}^{3/2}}}}$ converges.
