how to calculate the limit of $log(n)^{2/n}$ when $n$ goes to infinity?

Question

i need to use the Hadamard formula to find the convergence radius of a power series with complex values where $a_{n} = [\log(n)]^2$, i guess that $\displaystyle\sum a_{n}z^{n}$ diverges for all complex $z$ but if a should give a bet i would say that this limit goes to 1.

Answer 1

Take the log: $$\frac2n\log(\log n)\to 0\enspace\text{as}\enspace n\to\infty$$ since $\log n=o(n)$, $\log(\log n)=o(\log n)$ by substitution, hence $\log(\log n) = o(n)$ by transitivity.

Thus $\bigl(\log(\log n)\bigr)^{\tfrac 2n}\to 1$.