Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?
Solution 1:
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 $x \in 1_{\bigcup A_i}$.