Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|}n$ convergent？

Asked Dec 02 '14 at 09:32

Active Dec 02 '14 at 09:32

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It seems to be divergent to me because of this (i know that's not a proof just a thought):

$|sin(n)|\leq 1$ but for about half the values of n, $1/2<|sin(n)|$ for these values the series $ \sum\limits_{n=1}^\infty \frac{1/2}n$ diverges therefore $ \sum\limits_{n=1}^\infty \frac{|\sin n|}n$ diverges.

Can someone give me a hint or tell me if I'm at least going in the right direction.

Thank you so much

asked Dec 02 '14 at 09:32

wwbb90

1,121

The direction seems OK, but you are not there yet... – 5xum Dec 02 '14 at 09:35
You have $|\sin(n)|\geq (\sin(n))^2=\frac{1}{2}-\frac{\cos(2n)}{2}$ – Kelenner Dec 02 '14 at 09:35
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Hint: Each interval $[n\pi/2-1/2,n\pi/2+1/2]$, $n$ odd, contains an integer. – David Mitra Dec 02 '14 at 09:35
1

Or, see this. – David Mitra Dec 02 '14 at 09:37

Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|}n$ convergent？

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