The Ratio Test

(i) If   , then the series  is absolutely convergent

(and therefore convergent).

(ii) If   or  , then the series   is divergent.

(iii) If   , the Ratio Test is inconclusive; that is, no conclusion can be drawn about the convergence or divergence of   

EXAMPLE 4: Test the series   for absolute convergence.

We use the Ratio Test with   :

Thus, by the Ratio Test, the given series is absolutely convergent.

EXAMPLE 5:  Test the convergence of the series   .

Since the terms   are positive, we don’t need the absolute value signs.

Since  , the given series is divergent by the Ratio Test.

Note Although the Ratio Test works in Example 5, an easier method is to use the test for Divergence. Since  it follows that  does not approach 0 as  . Therefore the given series is divergent by the Test for Divergence.