To test the convergence of a series whose nth term is $a_{n}=\left({\frac{\log n}{\log(n+1)}}\right)^{n^2\log n}$ using Root test Test the convergence of the series whose nth term is given using Root test
$$a_{n}=\left({\frac{\log n}{\log(n+1)}}\right)^{n^2\log n}$$
I know that if $\lim_{n \to \infty}(a_{n})^{1/n}\lt 1$,the series converges
So,
let
$$ \lim_{n \to \infty}\left({\frac{\log n}{\log(n+1)}}\right)^{n\log n}=y$$
Taking log on both sides,
$$\lim_{n \to \infty}n\log n\log\left({\frac{\log n}{\log(n+1)}}\right)=\log y$$
But this is leading to complicated expressions.Is there a simpler way to approach this?

Hint
When $n$ is large
$$\frac{\log (n)}{\log (n+1)}\sim 1-\frac 1 {n \log(n)}$$
$$\log \left(\frac{\log (n)}{\log (n+1)}\right)\sim -\frac 1 {n \log(n)}$$
