Suppose I have two functions $f(x) = 1 - (1 + x)^{-m}$ and $g(x) = 1 - (1 + x)^{-n}$. And I use fixed point iteration to obtain a fixed point ($f(x) = x$ and $g(x) = x$).
How can I compare the rate convergence to fixed points between these two functions?
Assuming that the solution is $x$, the ratio
$$r:=\lim_{k\to\infty}\frac{x_{k+1}-x}{x_k-x}$$ gives you the speed of convergence (common ratio of an equivalent geometric sequence).
As $x_k$ tends to $x$, we get
$$r=\frac{1-(1+x_k)^{-m}-x}{x_k-x}=\frac{(1+x)^{-m}-(1+x_k)^{-m}}{x_k-x}\approx m(1+x)^{-m-1}=m\frac{1-x}{1+x}.$$
Unfortunately, the value of $x$ also depends on $m$ and we need an asymptotic analysis of this relation.
For large $m$, when $x\approx1$, we have
$$1-x\approx(1+1)^{-m}$$ and finally $$r\approx m2^{-m-1}.$$
This is decreasing, hence convergence improves with increasing $m$.