<s>[INST] Power Series Representation of $(1+x)/(1-x)$ For the power series representation of, $f(x) = \frac{1+x}{1-x}$ which is $1 + 2 \sum_{n=1}^\infty x^n$, Where does the added $1$ in front come from? How do I get to this answer from $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1}$
 [/INST] There are two ways to look at this. First, as you have noted, 
$$
\begin{align}
\frac{1+x}{1-x}
&=\sum_{n=0}^\infty x^n+\sum_{n=0}^\infty x^{n+1}\\
&=1+\sum_{n=1}^\infty x^n+\sum_{n=1}^\infty x^n\\
&=1+2\sum_{n=1}^\infty x^n
\end{align}
$$
The second is to notice that
$$
\begin{align}
\frac{1+x}{1-x}
&=1+\frac{2x}{1-x}\\[6pt]
&=1+2x\sum_{n=0}^\infty x^n\\
&=1+2\sum_{n=1}^\infty x^n
\end{align}
$$
</s>