convergence of a numerical series 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.

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