Find the power series representation of $\frac{1+x}{1-2x-x^2}$

Question

My textbook presents the following steps for finding the power series representation of $\frac{1+x}{1-2x-x^2}$:

$$f(x)=\frac{1+x}{1-2x-x^2}=\frac{\frac{1}{2}\alpha}{1-\alpha x} + \frac{\frac{1}{2}\beta}{1-\beta x}$$

I know that when dealing with this kind of problem, we usually try to do partial fraction decomposition. However, it seems that we cannot factor $1-2x-x^2$ here, so the textbook is using a different approach. So what is the technique used here?

Answer 1

Hint

An alternative way is$${1+x\over 1-2x-x^2}{={1+x\over 2-1-2x-x^2}\\={1+x\over 2-(1+x)^2}}$$now by defining $u\triangleq (1+x)^2$, what is the power series representation of ${1\over 2-u}$ in terms of $1+x$ and how is it related to the power series representation of $1+x\over 2-(1+x)^2$?

Answer 2

Hint: let $t=x+1$ then $${t\over 2-t^2}={t\over 2(1-({t\over \sqrt{2}}) ^2)}$$ $$= {t\over 2} \sum_{n\in\mathbb{N}} {({t\over \sqrt{2}}) ^{2n}}$$