If I have a function $f(z) = \sum a_k z^k$, and for example, then $e^{f(z)}$ should also has a power series expansion right? I can write $e^{f(z)} = \sum_i \frac{f(z)^i}{i!} = \sum_i \frac{1}{i!} (\sum_k a_k z^k)$.
My question is if I know the sum of the first 10 terms will give me error less than $0.1$, then what about the power series of $e^{f(z)}$ ? Is there any way I can analyze the convergence rate or radius of it? Thank you for any help.