I have this fraction: $\frac{(-12a^3)d^3 + (4wa^3 - 16a^2)d^2 + (5wa^2 - 8a)d - a^2w^2 + 2aw - 1}{(- 12wa^4 + 12a^3)d^3 + (4a^4w^2 - 20a^3w + 16a^2)d^2 + (4a^3w^2 - 11a^2w + 7a)d + a^2w^2 - 2aw + 1}$
How can I approximate it so it may be written as a function of increasing order of d? As in $f(d)+f(d^2)+H.O.T$
I have tried using Taylor series but that is centered around a point (which I don't want). I am looking into Pade approximation but am utterly confused. Can someone help me?
This problem is $$\frac{n_1d^3+n_2d^2+n_3d+n_4}{n_5d^3+n_6d^2+n_7d+n_8}$$ for complicated values of the constants $n_1,\ldots, n_8$.
Big O notation is useful in two contexts here; either for $d\to 0$ or for $d\to \infty$.
If you expand as $d\to 0$, you get $$\frac{n_4}{n_8}+d\frac{n_3n_8-n_7n_4}{n_8^2}+O(d^2)$$
If you expand as $d\to\infty$, you get $$\frac{n_1}{n_5}+\frac{1}{d}\frac{n_2n_5-n_6n_1}{n_5^2}+O\left(\frac{1}{d^2}\right)$$
You can calculate these easily (and more terms if desired) on alpha.