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.
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.