OK Let me thinking more
I cant apply A.S.T here since there is no (-1)^n or (-1)^(n-1) .. etc
Root and Ratio Failed since its algebric function
its not telescoping << am not sure about this .. maybe it needs some algebric operations to make it telescoping.
i have the comparison tests
but the problem i must prove that ( 1 - n*sin(1/n) ) is positive for all positive integers n > 1
Clearly n*sin(1/n) is positive
since n is positive and (1/n) are angles in the first quadrant for all n > 1
and sine is positive in the first quadrant
but the problem here i have ( 1 - n*sin(1/n) ) !!
if i proved that n*sin(1/n) < 1 for all n then i can use the comparison tests ..
but i cant prove it !
n*sin(1/n) ---> 1 n-->infinity
n*sin(1/n) = sin(1) as n=1
ohhh .. did i prove n*sin(1/n) belongs to [sin(1) , 1) for all n > 1
or this is wrong ?
Any mistakes here?