What are the values ​​for which the series converges? Determining the values ​​of $a$ and $b$ so that the series $\sum a_n$ converges, where 
 $$a_n=\ln n-a\ln(n+1)+b\ln(n+2)$$

\begin{align}
a_{n}
&=
\ln\left(n\right) - a\ln\left(n + 1\right) + b\ln\left(n + 2\right)
\\[3mm]&=
\ln\left(n\right)
-
a\left[
\ln\left(n\right) + {1 \over n} - {1 \over 2n^{2}} + \cdots
\right]
+
b
\left[
\ln\left(n\right) + {2 \over n} - {2 \over n^{2}} + \cdots
\right]
\\[3mm]&=
\left(1 - a + b\right)\ln\left(n\right)
+
\left(-a + 2b\right){1 \over n} + \left({1 \over 2}a - 2b\right){1 \over n^{2}} +
\cdots\,,
\qquad
n \gg 1
\\[5mm]&
\end{align}
$$
1 - a + b = 0\,,
\quad
- a + 2b = 0
\qquad\Longrightarrow\qquad
\color{#ff0000}{\large a = 2\,,\quad b = 1}
$$
