# How to determine the truncation error with products and quotients

If I have an equation given by

$$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$

and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is truncated at the $B^{th}$ order, $c$ is truncated at the $C^{th}$ order, and $d$ is truncated at the $D^{th}$ order, how do I go about determining the order of the truncation error for the full equation $Y$?

Given two truncated Laurent series, $f = p(x)+O(x^p)$, $g = q(x)+O(x^q)$, the orders will be $$f+g = p(x) + q(x) + O(x^{\min(p,q)}),$$ $$fg = p(x) q(x) + O(x^{\min(p+q_0, q+p_0)}),$$ $$1/g = 1/q(x) + O(x^{q-2q_0}).$$ where $p_0$ and $q_0$ are the degrees (possibly negative) of the first nonzero monomials in $p$ and $q$. So for each subexpression it is necessary to determine the leading term (which is possibly non-constant, and might be singular), then it is straightforward algebra.
• I follow you with the addition and multiplication, but I'm a bit confused on 1/g. Where does the 2 come from? Also...how does raising a term to a power influence the truncation error? Following your example, if it's $f^2$ does the truncation error become $O(x^{2p})$ or $O(x^{2+p})$? Could you possibly cater your answer to directly reflect the equation I presented? Given 4 truncated Laurent series... – ThatsRightJack Jul 7 '16 at 23:21