We know Maclaurin series expansion for many "standard" functions, like $e^x$, $\sin(x)$, $\cos(x)$, etc.
Often this expansion is used for approximating "hard" integrals or proving inequalities.
However, for what values is it true?
$$\sin(x)=\sum_{k=0}^\infty \frac{(-1)^k x^{2k+1}}{(2k+1)!}$$
This is Taylor series near $x=0$. Is it true when $x=2$?
If so, why?
There are also expansions near $\infty$. Wolframalpha says, that there is no expansion for sine near infinity. How are these expansions related and when to use an expansion near $\infty$?