To test convergence of infinite series $x^2(\log 2)^p + x^3(\log 3)^p + x^4(\log 4)^p +\dots$ 
To test convergence of infinite series $x^2(\log 2)^p + x^3(\log 3)^p + x^4(\log 4)^p +\dots$

My approach to the above problem:
let $u_n = x^{(n+1)}(\log(n+1))^p$. the $u_{(n+1)}= x^{(n+2)}(\log(n+2))^p$
now, $n \log\frac{u_n}{u_{n+1}}$=n\log($\frac{1}{x}$) + n$p$[\log(\log(n(1+$\frac{1}{n}$))-(\log(n(1+$\frac{2}{n}$)))]. After that I have used the expansion for $\log (1+x)$ and got stuck. 
It seems if I take the limit of $n \log\frac{u_n}{u_{n+1}}$, i.e $\lim_{n\to\infty}n \log\frac{u_n}{u_{n+1}}$ =$\infty$ $(>1)
$$\Rightarrow$The series seems to be convergent, which is not the answer. Please clarify the mistake and guide me.

There are a lot of possibilties to check convergence. 
E.g. the ratio test here for $p\ge 0$ is:
$$|\frac{x^{n+1}(\ln(n+1))^p}{x^n(\ln n)^p}|=|x|(\frac{\ln(n+1)}{\ln n})^p<1$$ 
This means $|x|<(\frac{\ln n}{\ln(n+1)})^p<1$: Convergence for $|x|<1$.
