Difference between $k[x,1/x]$ and $k(x)$

Question

What is the difference between $k[x,1/x]$ and $k(x)$ ?

Any rational polynomial with coefficients from $k$ belongs also to $k[x,1/x]$ or not?

Answer 1

No. You can't write $\frac1{x+1}$ as $\frac{a_{-n}}{x^n}+\frac{a_{-n+1}}{x^{n-1}}+\cdots+a_nx^n$.