convergence of a numerical series

Question

I would like to study the convergence of the numerical serie $$ S_n=\sum_{k= 1}^n u_k=\sum_{k= 1}^n \frac{1}{\left(\sqrt[k]{2}+\log k\right)^{k^2}}. $$ I tried the Cauchy rule (i.e. evaluate $\lim_{k\rightarrow +\infty}(u_k)^{\frac 1 k}$ but there is no issue.

Answer 1

$$\frac1{\left(\sqrt[k]2+\log k)\right)^{k^2}}<\frac1{\sqrt[k]{2}^{k^2}}$$

Answer 2

The series $$S=\sum_{k\geq 1}\frac{1}{\left(\sqrt[k]{2}+\log k\right)^{k^2}}$$ is convergent by comparison with a geometric series, since: $$ \left(\sqrt[k]{2}+\log k\right)^{k^2} \geq 2^k,$$ hence $S\leq 1$.