Let be a series of positive terms and suppose that , Then the following hold:
By definition, for given, we can choose such that In case , we start with such that . Then , i.e., .
Since, , the series is a convergent series. Thus by comparison test, is also convergent. In case, , we can start with such that . Then .
be a series of positive terms and suppose that 
,
, then the series is convergent. 
or, 
,the series is divergent. 
, the series may converge or diverge. 
given, we can choose 
such that 
, we start with 
such that 
. Then 
,
. 
, the series 
is a convergent series. Thus by comparison test, 
is also convergent. In case, 
, we can start with 
. Then 
.
