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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,

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  • $\begingroup$ What do you mean by bounding $|E(x)|$? It will go to infinity as $x$ does. $\endgroup$ Oct 15, 2014 at 8:45

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