Test the following series for convergence:
The series is convergent if and divergent if .
Proof. We will use the limit comparison test with series (which we know converges for and diverges for ). First, we rewrite the terms in the series,
Now, we can use the limit comparison test with .
(In the second to last line, we multiplied the numerator and denominator by and in the denominator we had and split these two pieces between the two terms in the product of the denominator.) Therefore, the given series converges or diverges as does. Since converges for and diverges for , we have proved our claim