Understanding the field of fractions of $F[[x]]$ (the ring of formal power series in the indeterminate x with coefficents in F) Let $F[[x]]$ be the ring of formal power series in the indeterminate x with coefficients in F. Show that the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series. 
I've been going in circles on this one for what seems like ever now! If anyone could lend me some insight I would be exceedingly grateful! Thanks!

$\sum_{k=-n}^\infty a_kx^k=\frac{\sum_{k=0}^\infty a_{k-n}x^k}{x^n}$, so every Laurent series can be written as a quotient of power series. 
Conversely, given a fraction $\frac{F(x)}{G(x)}$, write $G(x)=a_nx^n(1+xH(x)),$ where $H(x)$ is a power series. Then
$$
\frac1{G(x)}=\frac1{a_nx^n}\cdot \frac{1}{1+xH(x)}=\frac1{a_nx^n}\sum_{k=0}^\infty(-1)^kx^kH(x)^k
$$
When $H(x)^k$ is expanded into powers of $x$, and like terms are collected, the above writes $\frac1{G(x)}$ as a Laurent series. Since Laurant series are closed under multiplication, $F(x)\cdot \frac1{G(x)}$ is also a Laurent series.
