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### Section 10.4 : Convergence/Divergence of Series

2. Compute the first 3 terms in the sequence of partial sums for the following series.

$\sum\limits_{n = 3}^\infty {\frac{{2n}}{{n + 2}}}$ Show Solution

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*}}