Answer
Diverges
Work Step by Step
$\lim _{k\rightarrow \infty }\dfrac {\sqrt {k}}{\ln ^{10}k}=\dfrac {\dfrac {\partial }{\partial k}\sqrt {k}}{\dfrac {\partial }{\partial k}\ln ^{10}k}=\dfrac {\dfrac {1}{2\sqrt {k}}}{\ln ^{9}k\times \dfrac {1}{k}}=\dfrac {\sqrt {k}}{2\ln ^{9}k}=\dfrac {\dfrac {\partial }{\partial k}\sqrt {k}}{\dfrac {\partial }{\partial k}\left( 2\ln ^{9}k\right) }...=\dfrac {\sqrt {k}}{2^{10}}=\infty \neq 0$ then the series diverges