I'm trying to find out more and especially get a better idea about the background of a theorem that goes as follows:
$\underline{\text{Theorem}}$
If we have a radius of convergence $R>0$ of a series $$A(z) = \sum_{n=0}^{\infty}a_nz^n$$,
then for every $N \in \mathbb{N}$ and for every $0<r<R$ there is a number $c=c_{N+1}(r)$ such that $$|A(z) - \sum_{n=0}^{N}a_nz^n|\leq c|z|^{N+1}$$ for all z with $|z|\leq r$.
So if anybody might have a comprehensible reference or an explanation were this is "coming from", it would be very helpful.