We shall denoting by , the set of all such series whose terms are positive. Thus, is series of positive terms if and only if . If then its sequence or partial sums is monotonically increasing for . This gives

** Theorem:** If then either converges to positive number or diverges to .

** Proof:** If and its sequence of partial sums be bounded, such that . Thus being monotonic and bounded is convergent and . If be bounded, then being monotonic for diverges to , i.e. diverges to . Hence the theorem proved.

When the direct investigation regarding the boundedness of are not easy then we use alternative tests to establish the convergence of the series. In establishing various tests, given as

- If and such that .
- If and such that .

The above two results follows from the fact that all except a finite number of terms of the sequence lie in the neighborhood of the limit ‘*a*’.