How do you expand, say, $\frac{1}{1+x}$ at $x=\infty$? (or for those nit-pickers, as $x\rightarrow\infty$. I know it doesn't strictly make sense to say "at infinity", but I think it is standard to say it anyway).
I have a couple of interesting questions to follow... I might as well say them now.
Question 1. According to WolframAlpha, the Taylor expansion of, say, $\frac{1}{(1+x-3x^{2}+x^{3})}$ at $x=\infty$ is $\frac{1}{x^{3}}+\frac{3}{x^{4}}+\frac{8}{x^{5}}+...$ . We see that the expansion starts at $\frac{1}{x^{3}}$ and has higher order terms. I suspect this occurs for any fraction of the form 1/(polynomial in x). Why is this? (I don't see how dividing all the terms on the LHS by $\frac{1}{x^{3}}$ helps, for example).
Question 2. My motivation behind all this Taylor series stuff was originally: Can an infinite expansion $\frac{1}{a_{0}+a_{1}x+a_{1}x^{2}+...}$ be written in the form $b_{0}+\frac{b_{1}}{x}+\frac{b_{2}}{x^{2}}+...$ ? If so, when (i.e. what conditions must we have on the $a_{n}$)?