Answer
Diverges
Work Step by Step
Apply the ratio test to the series
$\Sigma^{\infty}_{n=1} (\frac{3n+2}{n+3})^n $
$\lim\limits_{n \to \infty} \sqrt[n] {|(\frac{3n+2}{n+3})^n|}$, Absolute value falls out because
$\lim\limits_{n \to \infty} \frac{3n+2}{n+3} = 3 >1$
Because the answer is greater than 1, the series diverges
