<s>[INST] What is the sum of the power series below? For $$\sum_{n=1}^{\infty}\frac{(n+2)}{n(n+1)}x^n$$
What is the sum of it?
 [/INST] Since 
\begin{align}
\frac{n+2}{n(n+1)} = \frac{2}{n} - \frac{1}{n+1}
\end{align}
then the series is as follows.
\begin{align}
\sum_{n=1}^{\infty} \frac{(n+2) \ x^{n} }{ n(n+1)} &= 2 \sum_{n=1}^{\infty} \frac{x^{n}}{n} - \sum_{n=2}^{\infty} \frac{x^{n-1}}{n} \\
&= - 2 \ln(1-x) - \frac{1}{x} \left( - \ln(1-x) - x \right) \\
&= \left( \frac{1}{x} - 2 \right) \ln(1-x) + 1.
\end{align}
</s>