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 $(c-r,c+r)$ ?

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Answer 1 · 2014-09-20T13:56:11

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 $(c-r,c+r)$ .

since $x$ in $(c-r,c+r)$ ,

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

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

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

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