F(x)=sinx with taylor's theorem

[tomboi03]

i. Expand f(x)= x^5+x^3+x in powers of (x-1), using Taylor's theorem
ii. Consider f(x)= sin x. Find the Taylor polynomial T of degree 7 expanded at $$\pi\$$/3.
Give an estimate for the remainder term, in the form
|sin (x)- T(x)|$$\leq$$ C|x-$$\pi$$/3|^8
with a suitable (good) constant C.

f(x)= f(1)+f'(1)(x-1)+f''(1)(x-1)^2... + f^(n)(1)/n!(x-1)^(n)
is that right for i?
and for ii... I'm not sure what to do.

 

[CompuChip]

For i, it is technically correct, but you are given the function. So work out the derivatives!

For ii, start by writing down the Taylor series, for n running up to 7.