Explaining and using the $N$-term Taylor series for $\sin x$ So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer:

Explain why the Taylor series containing $N$ terms is:
  $$\sin x = \sum_{k=0}^{N-1} \frac{(-1)^k}{(2k+1)!}x^{2k+1}+r_{2N-1}(x)$$
  with a remainder $r_{2N-1}(x)$ that satisfies:
  $$|r_{2N-1}(x)| \leqslant \frac{|x|^{2N}}{(2N)!}$$
How many terms of the series do you need to include if you want to compute $\sin x$ with an error of at most $10^{-3}$ for all $x \in [-\pi/2, \pi/2]$?
Compute $\sin(\pi/2)$ from the Taylor Series. How large is the actual error?'

I'm fairly certain I can do the third part, but the first and second have completely thrown me. Can anybody help, or at least point me in the right direction?
Cheers.
 A: The first part is a special case of a standard error estimate for alternating series (see e.g. here) and is proven in any introductory book on analysis. Here is an online proof for this and for other error estimates of truncated series: http://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx (see the section on the alternating series test).
For the second part, you just need to notice that the upper bound for the error term is maximised at the end points of the interval. So as long as you arrange for $r_{2N-1}(\pi/2)$ to be smaller than $10^{-3}$, the same will hold for all the other $x$ in the interval.
A: That is the general form of Taylor series (with remainder). There are several forms of the remainder, like Lagrange's. For the series centered at $a$:
$\begin{equation*}
  r_{N + 1}(x)
    = \frac{f^{(N + 1)}(\xi)}{(N + 1)!} (x - a)
\end{equation*}$
Here $\lvert \xi - a \rvert < \lvert x - a \rvert$ (i.e., $\xi$ is somewhere between $a$ and $x$).
In this case, the derivatives of $\sin x$ are always at most 1 in absolute value (for a shorter range you can get sharper bounds). As the series is alternating, the sign of the last term tells you in which direction the error goes.
