I'm not sure whether the following series converges or diverges:
∞∑n=1|sin(n)|n
I proved that the series ∑∞n=1sin2(n)n converge. Is there a way I can use that? I've tried using Dirichlet series test with the latter but didn't got nowhere since 1|sinx| is not monotone decreasing.
Answer
sin2(n)n=12n−cos(2n)2n
By Dirichlet's test, ∑cos(2n)2n converges, hence ∑sin2(n)n diverges (since ∑1n diverges to ∞).
sin2(n)n≤|sin(n)|n
So by the comparision test, ∑|sin(n)|n diverges
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