The sum diverges (integral test). We have
[tex]\displaystyle \sum_{n=2}^\infty \frac3{2n} \ge \int_2^\infty \frac3{2x}\,\mathrm dx[/tex]
and the integral diverges, so the sum also diverges.
[tex]\displaystyle\int_2^\infty\frac3{2x}\,\mathrm dx = \lim_{x\to\infty}\frac32\ln|x| - \frac32\ln(2) = \infty[/tex]
