6. Let $f_{1}(x)=\frac{2 x-1}{x+1}$, for $n \geqslant 2$, define $f_{n}(x)=f_{1}\left(f_{n-1}(x)\right)$.
If $f_{29}(x)=\frac{1+x}{2-x}$, then $f_{2009}(x)=$ $\qquad$
6. $\frac{1+x}{2-x}$.

Since $f_{30}(x)=f\left(\frac{1+x}{2-x}\right)=x$, we have,
$$
f_{31}(x)=f_{1}(x) \text {. }
$$

And $2009=30 \times 66+29$, thus,
$$
f_{2009}(x)=f_{29}(x)=\frac{1+x}{2-x} .
$$