Remember that \(n\)^th term in the sequence of partial sums is just the sum of the first \(n\) terms of the series. So, computing the first three terms in the sequence of partial sums is pretty simple to do.

Here is the work for this problem.

\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}{s_3} & = \frac{{2\left( 3 \right)}}{{3 + 2}} = \frac{6}{5}\\ {s_4} &= \frac{{2\left( 3 \right)}}{{3 + 2}} + \frac{{2\left( 4 \right)}}{{4 + 2}} = \frac{{38}}{{15}}\\ {s_5} & = \frac{{2\left( 3 \right)}}{{3 + 2}} + \frac{{2\left( 4 \right)}}{{4 + 2}} + \frac{{2\left( 5 \right)}}{{5 + 2}} = \frac{{416}}{{105}}\end{align*}}\]