Answer
Diverges
Work Step by Step
Given $$\sum_{n=1}^{n=\infty} \frac{1}{n-\ln n}$$
Compare with divergent series $\displaystyle\sum_{n=1}^{n=\infty} \frac{1}{n}$ ($p-$series $p=1$), by using the Limit Comparison Test
\begin{align*}
\lim_{n\to\infty} \frac{a_n}{b_n} &=\lim_{n\to\infty} \frac{ \frac{1}{n-\ln n}}{\frac{1}{n}}\\
&=\lim_{n\to \infty} \frac{n}{n-\ln n}\\
&=\lim_{n\to \infty} \frac{1}{1-\frac{1}{n} }\\
&=1
\end{align*}
Then the given series also diverges.
