How large should............?

Question

thanks to all who've answered me so far. i've asked this before but don't fully understand.

Let p(x) be the taylor poly. for f(x) = log(1-x) about a=0

How large should n be chosen to have |f(x) -p(x) <= 10^-4

for a) [-1/2,1/2]

b) [-1,1/2]

So I know that R(x) = (x-a)^n+1 / (n+1)! * f^(n+1)

But how do I get f^(n+1) I mean, which derivative do I use, and why? Thanks

Answer 1

|R(x)| = | log(1-x)^n+1 / (n+1)!  * x^(n+1) |

Plug in 1/2

f^(n+1) = the nth derivative. You can just choose, for example; try the 4th derivative, and n = 3

ie) 4th derivative = -6 / (x-1)^4

= -6 / (-1/2)^4 = 96

R(x) = 96* (1/2)^4 / 4! = 0.25

So is 0.25 less than 10^-4? No. So keep increasing n that way.