Theorem (Integral Test):

Let be a positive continuous decreasing function with
            
Then either both
            
converge or diverge.

For , consider the interval with the partition
          .
Then, since is decreasing,
         
Thus, if
         
then
                       ---------(*)
In case is convergent, we have for
        
Since is positive, is monotonically increasing and hence it is convergent. Conversely, if exists, then by the Sandwich theorem, (*) implies that is convergent.