OK Let me thinking more

N.T.T Failed

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?