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Series

Series

At a Glance - The Divergence Test

Since an uncontrolled grilled cheese spill is a hazardous materials catastrophe we'd like to avoid, we want a test that will tell us when not to open the Pandora's box.

We know that if a series converges, its terms must approach zero. Rephrasing this in the contrapositive: if the terms of a series don't approach zero, the series diverges. This statement lets us look at a series and, if the terms don't approach zero, conclude that the series diverges.

Be Careful: We can't use this statement to conclude that a series converges. We can only use it to evaluate if a series diverges. That's why we call it the Divergence Test. If the terms do approach zero, there's hope that the series might converge, but we would need to use other tools to really draw that conclusion.

Sample Problem

Does the series

converge or diverge?

Answer.

Look at the limit of the terms an = n as n goes to ∞. Since

the series diverges.

Example 1

Does the series

converge or diverge?


Exercise 1

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series  converges.


Exercise 2

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series  diverges.


Exercise 3

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series  might converge.


Exercise 4

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series  converges.


Exercise 5

Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.

The series  diverges.


Exercise 6

Use the divergence test to determine if the following series MUST diverge.


Exercise 7

Use the divergence test to determine if the following series MUST diverge.


Exercise 8

Use the divergence test to determine if the following series MUST diverge.


Exercise 9

Use the divergence test to determine if the following series MUST diverge.


Exercise 10

Use the divergence test to determine if the following series MUST diverge.


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