Chapter 7: Q. 9 (page 631)
If you suspect that a series converges, explain why you would want to compare the series with a convergent series, using either the comparison test or the limit comparison test.
Short Answer
If the series is convergent, comparing it to divergent series will yield no results since the behaviour of the series is dependent on the behaviour of the series .
As a result, if the series converges, it must be compared to a convergent series.