Calculus/Limit Test for Convergence

Limit TestEdit

The first test for divergence is the limit test. The limit test essentially tells us whether or not the series is a candidate for being convergent. It is as follows:

Limit Test for Convergence
If a series   and if   the series must be divergent. If the limit is zero, the test is inconclusive and further analysis is needed.

This follows because if the summand does not approach zero, when   becomes very large,   will be close to the non-zero   and the series will start behaving like an arithmetic series; remember that arithmetic series never converge.

However, one should not misuse this test. This is a test for divergence and not convergence. A series fails this test if the limit of the summand is zero, not if it is some non-zero  . If the limit is zero, you will need to do other tests to conclude that the series is divergent or convergent.

Example 1Edit

Determine whether or not the series

 

is divergent or if the limit test fails.

SolutionEdit

Because

 

the limit is not zero and so the series is divergent by the limit test.

Example 2Edit

Determine whether or not the series

 

is divergent or if the limit test fails.

SolutionEdit

Because

 

the limit is not zero and so the series is divergent by the limit test.

Example 3Edit

Determine whether or not the series

 

is divergent or if the limit test fails.

SolutionEdit

Because

 

the limit is zero and so the test is inconclusive. Further analysis is needed to determine whether or not the series converges or diverges.