$$ \begin{align} \sum_{n=2}^\infty (\log n)^n(z+1)^{n^2} \end{align} $$
What is the radius of convergence of this power-series?
I tried applying the root test and the ratio test , but I couldn't come to a conclusion.
Also, the $n^2$ in $(z+1)^{n^2}$ confuses me as to the behaviour of the series.
Essentially, we have to investigate the behaviour of the series $$S=\sum_n (\log n\cdot r^n)^n,$$ where $r$ is a real number. Notice that if $|r|\lt 1$, then $\log n\cdot r^n$ is smaller than $1/2$ for $n$ large enough, hence the series $S$ is convergent. If $r=1$, the series diverges.