Question

Find the sum of series:​

Answer 1

The sum diverges (integral test). We have

$\displaystyle \sum_{n=2}^\infty \frac3{2n} \ge \int_2^\infty \frac3{2x}\,\mathrm dx$

and the integral diverges, so the sum also diverges.

$\displaystyle\int_2^\infty\frac3{2x}\,\mathrm dx = \lim_{x\to\infty}\frac32\ln|x| - \frac32\ln(2) = \infty$