trying to find the power series expansion of $\frac{z^2}{1-z}$ around the point $z_0=0$ and having some trouble doing so. So far I've tried the following idea:
Let $\frac{z^2}{1-z} = z^2(\frac{1}{1-z})$. Then the expansion of $z^2$ is $\sum_{n=2}^2 z^n$ (???) and the expansion of $\frac{1}{1-z}$ is $\sum_{n=0}^\infty z^n$. But I'm not sure if this is valid, or how to evaluate the product.
Am I on the right track, or should I be going about this differently?
Best Answer
You’re over-complicating things. You have the main piece, that $$\frac1{1-z}=\sum_{n\ge 0}z^n\;;$$ now just multiply by $z^2$ to get $$\frac{z^2}{1-z}=z^2\sum_{n\ge 0}z^n=\sum_{n\ge 0}z^{n+2}=\sum_{n\ge 2}z^n\;.$$