<s>[INST] Summing $\sum_{n=1}^{\infty} \sin\frac{n!\pi}{120}$ How do I sum $$\sum_{n=1}^{\infty} \sin\frac{n!\pi}{120}$$ 
 [/INST] Note that $\sin \left(\dfrac{n! \pi}{120} \right) = 0$ for all $n \geq 5$. Hence, $$\sum_{n=1}^{\infty} \sin \left(\dfrac{n! \pi}{120} \right) = \sin \left(\dfrac{1! \pi}{120} \right) + \sin \left(\dfrac{2! \pi}{120} \right) + \sin \left(\dfrac{3! \pi}{120} \right) + \sin \left(\dfrac{4! \pi}{120} \right)$$
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