# How do you find the largest interval (c-r,c+r) on which the Taylor Polynomial p_n(x,c) approximates a function y=f(x) to within a given error?

Oct 6, 2014

Let us assume that there is $M > 0$ such that

$| {f}^{\left(n + 1\right)} \left(x\right) | \le M$ for all $x$.

If we want the error to be less than $\epsilon > 0$, then

|R_n(x;c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|,

where $z$ is between $x$ and $c$.

by replacing $| {f}^{\left(n + 1\right)} \left(z\right) |$ by $M$,

le M/{(n+1)!}|x-c|^{n+1} < epsilon

By solving for $| x - c |$,

Rightarrow |x-c| < root(n+1){{epsilon (n+1)!}/M}

Hence,

r=root(n+1){{epsilon (n+1)!}/M}

I hope that this was helpful.