Could someone help me with this problem? Thanks.
"Prove that the Taylor series of sin(x) at a=pi/2 represents sin(x) for all x."
I already know how to find the Taylor series, but I have no idea how to prove anything.
Use Taylors' remainder theorem
Taylor's theorem - Wikipedia, the free encyclopedia
Note that for all $\displaystyle x \in \mathbb{R}$
$\displaystyle |\sin(x)|\le 1$ and $\displaystyle |\cos(x)|\le 1$. Also that for each x that the remainder (the error) goes to $\displaystyle 0 \text{ as } n \to \infty$
I think you are mistaken. Both $\displaystyle \sin(x)$ and $\displaystyle \cos(x)$ are real analytic entire functions.
Consider the remainder term from the link above it gives
$\displaystyle \displaystyle R_n(x)=\frac{f^{n+1}(c)}{(n+1)!}\left(x-\frac{\pi}{2} \right)^{n+1}$
Since all derivatives of $\displaystyle \sin(x)$ are less than 1
$\displaystyle \displaystyle |R_n(x)| \le \frac{1}{(n+1)!}\left|x-\frac{\pi}{2} \right|^{n+1} $
now for any $\displaystyle y \in \mathbb{R}$ we can choose an $\displaystyle n$ so large that the remainder is as small as we wish
$\displaystyle \displaystyle |R_n(y)| \le \frac{1}{(n+1)!}\left|y-\frac{\pi}{2} \right|^{n+1} < \epsilon$