Answer
The radius of converges is $R=1$ and converges for $-1\leq x\lt 1$.
Work Step by Step
Apply the ratio test:
\begin{aligned}
\rho&=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right|\\
& =\lim _{n \rightarrow \infty}\left|\frac{\frac{x \times x^{n}}{(n+1)-4 \ln (n+1)}}{\frac{x^{n}}{n-4 \ln n}}\right|\\
&=\lim _{n \rightarrow \infty}\left|x \cdot \frac{n-4 \ln n}{(n+1)-4 \ln (n+1)}\right|\\
& =|x| \times \lim _{n \rightarrow \infty} \frac{n-4 \ln n}{(n+1)-4 \ln (n+1)}\\
&= |x| \times \lim _{n \rightarrow \infty} \frac{1-\frac{4}{n}}{1-\frac{4}{n+1}}
\end{aligned}
Then the radius of converges is $R=1$ and the series converges for $-1\lt x\lt 1$.
For $x=1 $, the series $\sum_{n=9}^{\infty} \frac{1}{n-4 \ln n} $ diverges by using the comparison test with $\sum_{n=9}^{\infty} \frac{1}{n } $.
For $x=-1 $, the series $\sum_{n=9}^{\infty} \frac{(-1)^n}{n-4 \ln n} $ is an alternating series; when n increases, the value of $a_{n}$ decreases and $\lim _{n \rightarrow \infty}\left|a_{n}\right|=0$. So, the using Alternating Series Test, we can say that the series converges.
