Answer
The series $\sum_{n=1}^{\infty} \frac{1}{(\ln n)^4}$ diverges.
Work Step by Step
Since $\ln n\lt n^{1/4}$, then $(\ln n)^4\lt n $ and hence
$$\frac{1}{(\ln n)^4}\gt \frac{1}{n}.$$
Now, since the series $\sum_{n=1}^{\infty} \frac{1}{n} $ is a divergent p-series, then the series $\sum_{n=1}^{\infty} \frac{1}{(\ln n)^4}$ diverges.
