$$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}n)}n$$
Using the comparison test/limit comparison test? I have tried the comparison test and several attempts at the limit comparison test, but everything I try points to divergence, which I know isn't true.
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You might be able to do something with Dirichlet's Test – Davis Yoshida Mar 05 '14 at 05:52
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1Compare with one half of the harmonic series, i.e., consider those n where $\cos\tfrac1n>\tfrac12$. And please give the reasons that make you exclude divergence. – Lutz Lehmann Mar 05 '14 at 05:55
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Sorry, I meant sin, not cos! – rosstex Mar 05 '14 at 05:56
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Just compare with $1/n^2$. – Mariano Suárez-Álvarez Mar 05 '14 at 05:56
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$|\sin(x)|\le \min(1,|x|)$. – Lutz Lehmann Mar 05 '14 at 05:57
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Mariano, could I not compare $1/n$ to $1/n^2$ by that logic? – rosstex Mar 05 '14 at 05:57
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Not at all. Have you tried computing the limit of the quotient of the general term of your series with $1/n^2$? (Or the inequality $0\leq \tfrac1n\sin\tfrac1n\leq\tfrac1{n^2}$. – Mariano Suárez-Álvarez Mar 05 '14 at 05:59
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LutzL, my reasons only include WolframAlpha – rosstex Mar 05 '14 at 05:59
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The question for reasons was from before the change from $\cos$ to $\sin$. – Lutz Lehmann Mar 05 '14 at 06:14
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Just for your information (hoping that this could be of any interest) : the infinite sum is $1.47283$ while the infinite sum of $1/n^2$ is $\pi^2/6=1.64493$. – Claude Leibovici Mar 05 '14 at 10:00
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The series is convergent:
$$ 0\le \frac{\sin\frac 1n}{n} \le \frac{\frac 1n}{n} = \frac 1{n^2} $$ and $\sum \frac 1{n^2} <\infty$.
mookid
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Thanks! Would you mind pointing out the error in my logic of using the limit comparison test with $b_n=\frac1n$ and using L'Hôpital's? – rosstex Mar 05 '14 at 06:08
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@rosstex, edti the question to include your reasoning. It is not possible to point out the error in what you did if you do not tell us, well, what you did! – Mariano Suárez-Álvarez Mar 05 '14 at 06:09
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@MarianoSuárez-Alvarez Sorry, I'm just getting used to Latex! You guys are very helpful, I'm on it – rosstex Mar 05 '14 at 06:10
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