When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way?
Also, is there an elementary reason why the Taylor series approximates a function? I've read the wikipedia entry but didn't understand it so it's greatly appreciated if someone can point me in the right direction.
This is not a complete answer, but alternating sums sometimes can be factored into products of form $\prod_{i=1}^k (1-a_i)$, which is 0 when any $a_i$ is 1.
Sine factored:
$\sin \pi x = \pi x \prod_{i=1}^{\infty} (1-x^2/n^2)$, which is $0$ when $x \in \mathbb Z$.
The inclusion-exclusion principle factored:
$\prod_{i=1}^{k} (1 - 1_{A_i}(x))$, which is $0$ when $x \in 1_{\bigcup A_i}$.