Comparison Test

Assume that we have two types of series ∑a_n and ∑b_n with a_n, b_n ≥ 0 for all n and a_n ≤ b_n for all n. 

Then,

A.  If ∑b_n is convergent then this is ∑a_n.

B.  If ∑a_n is divergent then this is ∑b_n.

Alternatively, we have two series of positive terms and the terms of one of the series are all time larger than the terms of the other series. After that if the larger series is convergent the smaller series must as well be convergent.  Similarly, if the smaller series is divergent as compared to the larger series must as well be divergent. 

Note: That in order to apply this test we require both series to start at similar place.