Answer
converges
Work Step by Step
Given $$\sum_{n=1}^{\infty}(\ln n-\ln (n+1))$$
Since
\begin{align*}
a_n &= \ln n-\ln (n+1)\\
&=\ln \left(\frac{n}{n+1}\right)\\
&=\ln \left(\frac{1}{1+\frac{1}{n}}\right)
\end{align*}
Then $$\lim_{n\to\infty } a_n = \lim_{n\to\infty } \ln \left(\frac{1}{1+\frac{1}{n}}\right)=0 $$
Hence, the series converges.
