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To make the relationship between $x$ and $X$ explicit, in the sense that we want $|x/X|\ll1$ instead of simply assuming that $x$ is small, we can set $z=x/X$ and series-expand for small $z$: F = Sum[a[i] * x^i * X^(7 - i), {i, 6}]; Series[F /. x -> z*X, {z, 0, 3}] (* a[1] X^7 z + a[2] X^7 z^2 + a[3] X^7 z^3 + O[z]^4 *) In practice this is pretty ...

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If you only need the terms up to a given order in one of the polynomial's variables, you could just use Series and Normal: Series[expr, {x, 0, 3}] Normal[%] Series returns a result in terms of SeriesData objects, which include O[x^n] terms that allow Mathematica to keep track of the validity of the approximation. Normal takes a SeriesData object and ...

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I'm not sure what is meant by a two-point Padé approximant. I'm not familiar with multipoint Padé approximants, and there seem to be a variety of types. One definition is that the series expansion of the rational approximant should agree with the series of expansion of the function to be approximated at each point to individually specified orders. This ...

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Here is an idea how you could proceed: Let's take your example of f[x_]=Exp[x] and you would like a 2 point approx. at x=0 and x=1. Toward this aim, you could make a Pade approximation at x=0 and x=1 with a numerator degree of m (e.g.2) and denominator degree of n (e.g. 2). This gives you two rational functions, one for x=0 and one for x=1. Now, you could ...

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