Since diverges, also diverges by the Limit Comparison Test.
THE RATIO TEST
Let be a nonnegative series, and let , if it exists. Then
converges if L < 1 and diverges if L > 1.
If L = 1, this test is inconclusive; apply one of the other tests.
NOTE: It is good practice, when using the ratio test, to first write ; then,
if it is known that the ratio is always nonnegative, you may rewrite the limit
without the absolute value. However, when using the ratio test on a power series
(see Examples 33–36), you must retain the absolute value throughout the limit
process because it could be possible that x < 0.
Example 22 __
Does , converge or diverge?
SOLUTION:
Therefore this series converges by the Ratio Test.
BC ONLYExample 23 __
Does converge or diverge?
SOLUTION:
and
(See §E2, Chapter 1.) Since e > 1, diverges by the Ratio Test.
Example 24 __
If the Ratio Test is applied to any p-series, , then