The series $$\(\sum_{n=0}^{\infty} \frac{|\sin x|^{n}}{n!}\)$$ I. converges because $$\(\frac{|\sin x|^{n}}{n!} \leq \frac{1}{n!}\)$$. II. converges because $$\(\sum_{n=0}^{\infty} \frac{1}{n!}=e\)$$.

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