Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials):
(a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$;
(b) the first $k$ coefficients of its Taylor expansion, say $c_1,...,c_k$.
Using the $k$ given coefficients, we can build a Padè approximant $A(x)$ such that the error $E(x)=P(x)-A(x)$ is $O(x^{k+1})$ for $x\rightarrow 0$.
My question: is there any explicit way of bounding the truncation error $|E(x)|$, from the given data? By 'explicit', I do not necessarily mean a closed formula. Any reference on the subject would be much appreciated.
Regards,
Michele