Wednesday, 25 April 2018

real analysis - Does suminftyn=1fracleft|sinleft(nright)right|n converge?



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=12ncos(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


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...