Taylor/Maclaurin series

[dangerous_dave]

I have quite a few questions relating to these. The first is:

find the tdegree 8 taylor polynomial p(x) of f(x) = sin x about c= pi/2

Estimate the error in p(2) to approximate sin (2).

I am having trouble finding anything that helps with this. And anything I did find I couldn't understand.

Please help.

 

[skeeter]

general form for an nth degree Taylor polynomial for f(x) centered at x = a ...

\(\displaystyle f(x) \approx P(x) = f(a) + f'(a)(x-a) + f''(a)\frac{(x-a)^2}{2!} + f'''(a) \frac{(x-a)^3}{3!} + ... + f^n(a) \frac{(x-a)^n}{n!}\)

the "a" value is pi/2.

\(\displaystyle f(x) = \sin(x)\) ... \(\displaystyle f\left(\frac{\pi}{2}\right) = 1\)

\(\displaystyle f'(x) = \cos(x)\) ... \(\displaystyle f'\left(\frac{\pi}{2}\right) = 0\)

\(\displaystyle f''(x) = -\sin(x)\) ... \(\displaystyle f''\left(\frac{\pi}{2}\right) = -1\)

\(\displaystyle f'''(x) = -\cos(x)\) ... \(\displaystyle f'''\left(\frac{\pi}{2}\right) = 0\)

see a pattern here? so what would the 8th degree Taylor polynomial for f(x) = sin(x) be?

note that the terms of the Taylor polynomial are alternating in sign ... what do you know about the error bound for alternating series?