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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 $G_n$ ?

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

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