# How do you use the Taylor Remainder term to estimate the error in approximating a function y=f(x) on a given interval (c-r,c+r)?

Oct 16, 2014

Assume that there exists a finite $M > 0$ such that

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

for all $x$ in $\left(c - r , c + r\right)$.

The error of approximating $f \left(x\right)$ by the Taylor polynomial p_n(x;c) can be estimated by

|f(x)-p_n(x;c)|

=|R_n(x;c)|

=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|, where $z$ is between $x$ and $c$

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

I hope that this was helpful.