Convergence/Divergence of $\sum_{n=1}^{\infty} \sin(1/n)$

it is a question Convergence/Divergence of calculus II! Please give me a hand!

Determine convergence or divergence using any method covered so far.

$$\sum_{n = 1}^{\infty} \sin (1/n)$$

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This looks like homework. What have you tried so far? Also, most of us probably don't know "methods covered so far", in your class. Perhaps you would care to state some of the things you are expected to use? – Aryabhata Nov 11 '10 at 17:20

You need to determine convergence for $\sum_{n=1}^\infty\sin(1/n)$. The series diverges. The two hints below may guide you when trying to justify this.

Hint 1: $\lim_{\theta\to0}\sin(\theta)/\theta=1$ and $1/n\to 0$ as $n\to\infty$.

Hint 2: $\sin(1/n)$ is positive. So you may attempt a limit comparison test.

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We have some guidelines regarding homework questions: meta.math.stackexchange.com/questions/415/… and meta.math.stackexchange.com/questions/107/…. Disclaimer: Just making sure you are aware of the guidelines regarding homework. Not trying to force you to follow them. – Aryabhata Nov 11 '10 at 17:26
It wasn't marked as homework, but if you think it is better, I'll edit my answer accordingly. Let me know. (Edit: I re-read the question, and yes, it clearly looks like homework. Oh, well.) – Andrés E. Caicedo Nov 11 '10 at 17:35
The user just fired the question and left (typical of people who aren't really interested in learning). So we didn't get a chance to clarify. But the statement: "using methods covered so far" should make that clear I suppose. I suggest you edit your answer only if you agree with the guidelines. – Aryabhata Nov 11 '10 at 17:39
btw, the old title said Calc II (I had edited the title to make it more descriptive)! I forgot. I have added the old title to the question body. – Aryabhata Nov 11 '10 at 17:44

Hint: $\sin(x) / x \to 1$ as $x \downarrow 0$.

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