# Thread: Error Term Taylor Formula

1. ## Error Term Taylor Formula

Use the following form of the error term

$

R_{n+1}(x) = \frac{f^{n+1}(c)}{(n+1)!} x^{n+1}
$

where c is between 0 and x, to determine in advance the degree of Taylor polynomial at a = 0 that would achieve the indicated accuracy in the interval [0,x] for...

... $

f(x) = e^x, x = 2, error < 10^{-3}
$

(Do not compute Taylor polynomial.)

Thanks for the help!

2. Originally Posted by juicysharpie
Use the following form of the error term

$

R_{n+1}(x) = \frac{f^{n+1}(c)}{(n+1)!} x^{n+1}
$

where c is between 0 and x, to determine in advance the degree of Taylor polynomial at a = 0 that would achieve the indicated accuracy in the interval [0,x] for...

... $

f(x) = e^x, x = 2, error < 10^{-3}
$

(Do not compute Taylor polynomial.)

Thanks for the help!
Using your remainder formula with $f(x) = e^x$ on $[0,2]$

$

R_{n+1}(2) = \frac{e^c }{(n+1)!} 2^{n+1} \le \frac{e^2 }{(n+1)!} 2^{n+1} < \frac{9}{(n+1)!} 2^{n+1} \le 10^{-3}
$
which gives $n = 10$ although nine terms will work (trial and error).