Answer
$S_4\approx −0.183333$
$S_7 \approx 0.065079 $
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{n-2}{n^{2}+2 n}$$
Since
\begin{align*}
S_4 &= \sum_{n=1}^{4} \frac{n-2}{n^{2}+2 n}\\
&= \frac{-1}{3}+0+\frac{1}{15} +\frac{2}{24}\\
&\approx −0.183333
\end{align*}
and
\begin{align*}
S_7 &= \sum_{n=1}^{7} \frac{n-2}{n^{2}+2 n}\\
&= \frac{-1}{3}+0+\frac{1}{15} +\frac{2}{24} + \frac{3}{35}+\frac{4}{48}+\frac{5}{63}\\&\approx0.065079
\end{align*}
