So for example $\sin(x)$ can be written as product over the zeros:
$$x \prod\limits_{n=1}^{\infty} \left(1-\frac{x^2}{(n\pi)^2}\right)$$
and as a taylor series sum:
$$\sum\limits_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{(2n+1)!}$$
One could exapand out the first equation and compare it with the second. But to me it's not obvious that the second equation should result in a function with regular zeros at intervals of $n\pi$. Is there known some formal theory about telling what kind of patterns of zeros you'd get from particular taylor series just knowing the coefficients?