Calculus/Limit Test for Convergence

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Limit Test[edit]

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 1[edit]

Determine whether or not the series

is divergent or if the limit test fails.

Solution[edit]

Because

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

Example 2[edit]

Determine whether or not the series

is divergent or if the limit test fails.

Solution[edit]

Because

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

Example 3[edit]

Determine whether or not the series

is divergent or if the limit test fails.

Solution[edit]

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.