The idea with this test is that if each term of one series is smaller than another, then the sum of that series must be smaller. So, if every term of a series is smaller than the corresponding term of a converging series, the smaller series must also converge. And if a smaller series diverges, the larger one must also diverge.
As an example, consider the series
Compare that with a second series as follows:
(since n+1<2n for n>=1) .
Since this new, smaller sum diverges (it is a harmonic series), the original sum also diverges.
For another example, look at
Compare that with a second series also:
converges (since it is a p-series with p greater than one), so the first sum also converges.
If you have questions or comments, don't hestitate to contact us.