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I'm trying to understand how one would understand the error of a given Padé approximation for a function.

For instance, the $[2,1]$ approximant for $\log(1+x)$ is $\frac{x(6+x)}{6+4x}$. Is there a reasonable way to understand how good this approximation is for $|x|<1$ (without using the fact that we can actually go and calculate $\log(1+x)$ for any value we want)?

I understand there isn't a nice, closed form bound like what we have in Taylor's Theorem, but I suspect it is still possible to gain some understanding or bound to the theoretical error.

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Yes, there are formulas that give you an error term depending on the regularity of the function. They are particularly useful for diagonal Pade approximates. This survey http://arxiv.org/pdf/math/0609094v1.pdf provides some useful information.

In practice, one can simply expand both functions, say $\log (1+x)$ and $\frac{x(6+x)}{6+4x}$ into the Taylor series in $|x|\le 1$ and after "cancelling" the corresponding equal terms, you will get an error term. Alternatively, you can set up the function $f(x)=\log{(x+1)}- \frac{x(6+x)}{6+4x}$ and study it's behaviour using derivatives.

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