# Simplifying a polynomial by a nice recursive formula

Let define a function $g(x)= (1+x^2 )/2$ and then define again $G_i$ where $G_1(x) = g(x)$ and $G_{n+1}(x) = g(G_{n}(x))$ . How can we approximate $G_{2n}$ and $G_{3n}$ with respect to $G_n$ ? My idea was to write down the Taylor expansion but I don't know which temr's approximation would be useful to have a relation between $G_{kn}$ and $G_n$ ?

-

$G_{2n}(x)=G_n(G_n(x))$. So, you don't need an approximation.