I am trying to solve an exercise from the book "p-Adic Lie Groups" by Peter Schneider.
For any $\epsilon > 0$ the power serie $f(X)=\sum_{\mathbb{N}^r_0} X^\alpha\nu_\alpha$ is called $\epsilon$-convergent if $lim_{|\alpha|\to\infty} \epsilon^{|\alpha|}||\nu_\alpha||=0$. The K-vector space [even a Banach space] $\mathcal{F}_\epsilon(K^r;V) :=$ all $\epsilon$-convergent power series $f(X)=\sum_{\mathbb{N}^r_0} X^\alpha\nu_\alpha$ is normed by $||f||_\epsilon = max_{\alpha}$ $\epsilon^{|\alpha|}*||\nu_\alpha||$.
For the exercise let $g(X)=X^p-X\in\mathcal{F}_1(\mathbb{Q}_p;\mathbb{Q}_p)$ and $f(Y)=\sum_{n=0}^{\infty} Y^n \in \mathcal{F}_{1/p}(\mathbb{Q}_p;\mathbb{Q}_p)$. I want to show that $f \circ g = \sum_{n=0}^{\infty}$ $1/(1+X^p-X) \notin \mathcal{F}_1(\mathbb{Q}_p;\mathbb{Q}_p)$
It should be enough, that there exist infinitely many coefficents which are not divisible by p, but actually I do not know how I can show this fact. Thank you for any hints!
If there were only a finite number of coefficients that were not divisible by $p$, then after reducing everything modulo $p$, $f \circ g$ would be a polynomial, and then you would get $(f \circ g) \times (1 - X + X^p) = 1$. Looking at the degrees, $f \circ g$ must be a polynomial of degree $-p$, which is impossible.