# How do you find the smallest value of n for which the Taylor Polynomial p_n(x,c) to approximate a function y=f(x) to within a given error on a given interval (c-r,c+r)?

Sep 20, 2014

Let $\epsilon > 0$ be an acceptable error.

The error can be estimated by

|f(x)-p_n(x,c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|,
where $z$ in $\left(c - r , c + r\right)$.

since $x$ in $\left(c - r , c + r\right)$,

\leq |f^{(n+1)}(z)|/{(n+1)!}r^{n+1}

if we know that the absolute value of all derivatives of $f(x)$ are bounded by some fix value $M$, then

leq M/{(n+1)!}r^{n+1}

Find $n$ such that

M/{(n+1)!}r^{n+1} le epsilon