Consider a polynomial in $x$ $$ x\mapsto\sum_{n=0}^{N}a_{n}x^{n}. $$ Suppose the root of this polynomial is in $\left(0,1\right)$. Suppose further that we approximate the root of this polynomial by $\tilde{y}$, attained by dropping high order terms: $$ \sum_{n=0}^{M}a_{n}\tilde{y}^{n}=0\text{ where }M<N. $$ Then, letting the exact root be $y$, we have $$ 0=\sum_{n=0}^{N}a_{n}y^{n}-\sum_{n=0}^{M}a_{n}\tilde{y}^{n}=\sum_{n=0}^{M}a_{n}\left(y^{n}-\tilde{y}^{n}\right)+\mathcal{O}\left(y^{M+1}\right). $$ Can the error $\left|y-\tilde{y}\right|$ be bounded in some meaningful way?
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Are you saying the polynomial has exactly one root in $(0,1)$? The issues is that if you call this root $y$, then it doesn't make sense to say you're approximating with $\tilde{y}$ from the lower order sum because the lower order sum could have a bunch of roots in $(0,1)$, so which is your approximation? In other words, how are you discerning which $\tilde{y}$ you want? – Alex R. Feb 13 '15 at 20:20