Answer
Diverges.
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n}{10 n+12}$$
Since
\begin{align*}
\lim _{n \rightarrow \infty} \frac{n}{10 n+12}&=\lim _{n \rightarrow \infty}\left(\frac{1}{10+\frac{12}{n}}\right)\\
&=\frac{1}{10} \neq 0
\end{align*}
Since the $n$th term $a_{n}$ does not converge to zero, thus the series diverges.
