Which of the following statements about the series $\sum_{n=1}^{\infty} \sin(\frac{1}{n})$ is true? A The series diverges by the $n$th term test. B The series diverges by comparison to the series $\sum_{n=1}^{\infty} \frac{1}{n}$. C The series diverges by limit comparison to the series $\sum_{n=1}^{\infty} \frac{1}{n}$. D The series diverges by limit comparison to the series $\sum_{n=1}^{\infty} n$.
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First, we check the $n$th term test for divergence. We need to find the limit of the term $\sin(\frac{1}{n})$ as $n$ approaches infinity. $$\lim_{n \to \infty} \sin(\frac{1}{n}) = \sin(\lim_{n \to \infty} \frac{1}{n}) = \sin(0) = 0$$ Since the limit is 0, the Show more…
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