Now, for any such that


Hence, by the limit comparison test, the series
         
is also


Next, suppose the series is divergent for some . Let
       
Then for every such that

for if so then by the above discussion, it must converge for such that, not true. Hence, if the series diverges for some and , then it diverges for every with . Let


Note that, . If is bounded above, let


Then, the series converges absolutely for such that and diverges for such that . In case is not bounded, above, the series converges absolutely for every .