<s>[INST] Inverse of Power Series I am reading something that considers the following power series in $\mathbb{C}$:
$$ f(z) = z + a_2z^2 + a_3z^3 + a^4z^4 + ... \\ |z| < 1 $$
They define:
$$g(z) = z + b_0 + b_1z^{-1} + b_2z^{-2} + b_3z^{-3} + ... \\ |z| > 1 $$
And assert: $g(z) = \frac{1}{f(z^{-1})}$ and $f(z) = \frac{1}{g(z^{-1})}$. I'm having trouble seeing this though. Ideas?
 [/INST] By direct calculation, we see that
\begin{align}
f(z^{-1}) =&\ \frac{1}{z}+a_2\frac{1}{z^2}+a_3\frac{1}{z^3}+\ldots=\ \frac{1}{z}\left(1+a_2\frac{1}{z}+a_3\frac{1}{z^2}+\ldots \right)
\end{align}
which means
\begin{align}
\frac{1}{f(z^{-1})} =&\  \frac{z}{1+\left(a_2\frac{1}{z}+a_3\frac{1}{z^2}+\ldots  \right)} = z\sum^\infty_{n=0}(-1)^n\left( a_2\frac{1}{z}+a_3\frac{1}{z^2}+\ldots  \right)^n\\
=&\ z\left( 1+\sum^\infty_{k=1} b_{k-1}\frac{1}{z^k}\right).
\end{align}
</s>