The Limit Comparison Test

Suppose that   and     are series with positive terms. If

where c is a finite number and , then either both series converge or both diverge.

EXAMPLE 4: Test the series

for convergence or divergence.

We use the Limit Comparison Test with   and obtain 

Since this limit exists and   is a convergent geometric series, the given series converges by the Limit Comparison Test.

EXAMPLE 5:  Determine whether the series 

converges or diverges.

The dominant part of the numerator is   and the dominant part of the

denominator is  . This suggests taking

Since  is divergent  , the given series diverges by the Limit Comparison Test.

Notice that in testing many series we find a suitable comparison series   by keeping only the highest powers in the numerator and denominator.