Assume that f is a function with |f^(n)(x)|<or= 1 for all n and all real x. (The sine and cosine functions have this property.
Find the least integer n for which you can be sure that Pn(2) approximates f(2) within 0.001.
Thanks for the help!
Assume that f is a function with |f^(n)(x)|<or= 1 for all n and all real x. (The sine and cosine functions have this property.
Find the least integer n for which you can be sure that Pn(2) approximates f(2) within 0.001.
Thanks for the help!
Which form or the remainder are you using? Which point are you expanding about?
Expanding about $\displaystyle 0$, and using the Lagrange form of the remainder I get:
$\displaystyle |P_n(2) - f(2)| \leq \frac{2^{n+1}}{(n+1)!}$,
so the $\displaystyle n$ required is the smallest positive integral solution of:
$\displaystyle \frac{2^{n+1}}{(n+1)!}\le 0.001$
Now expanding about $\displaystyle 2$ you will requre only one term, that is $\displaystyle P_0(2)$ is exact when expansion is about $\displaystyle 2$.
RonL