Programmed tutorial: Nth-Term Test of Divergence: Example 1

Use the nth-Term Test to determine the divergence of these series:

General Contents

Detailed Contents

Index

Describe the nth-Term Test.

We test the limit of the n-th term of a series.

What condition on the limit do we use to determine the divergence of a series?

If , the series diverges.

What if the limit is 0?

In that case, we can not draw any conclusion about divergence or convergence of the series.

Plot the terms of
Then check your graph by clicking “Next”.

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What does the graph suggest about ?

It seems to be approaching 0.

Is this consistent with ?

Yes, the denominator keeps getting larger, so the terms get smaller and approach 0.

What can we conclude about the convergence or divergence of this series?

We can’t conclude anything with this method.

Now let’s examine the other series. 
Plot the terms of

Then check your graph by clicking “Next”.

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What does the graph suggest about ?

It appears to approach 0.35, approximately.

How can we determine ?

We can divide the numerator and denominator by n!

Do that.

We get

What value does the term approach?

It approaches 0 because its denominator keeps growing.

What then is the value of

Is this consistent with our graph?

Yes.

What can we conclude about the series

Since the limit of the nth term is not 0, we conclude that the series diverges.

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