Programmed tutorial: Absolute Convergence: Example 1

Let’s try to visualize this series.  Are the terms all positive?

No, the cosine function varies between –1 and 1.

To check whether this is an alternating series, determine the signs of the first two terms.

Since the first two terms involve cos(2) = -0.42 and
cos(4) = -0.65,
we see that the series is not alternating.

In order to visualize this series, plot its terms on paper.  Then check your plot by clicking “Next”.

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To get further insight, plot the sequence of partial sums on paper.  Then check by clicking “Next”.

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Does the series appear to converge?

Yes, it appears to converge to –0.45.

What method can we use here?

We can test for Absolute Convergence.

Describe this test in terms of a_n.

If converges, also converges.

What will our look like?

How does compare with ?

Since the numerator of is always ≤ 1,
we see that

What type of series is ?

It is a p-series.

What is the value of p?

p is equal to 3 here.

Is this series convergent or divergent?

Since p > 1, this p-series is convergent.

Summarize our analysis so far.

We have shown that the terms of
are smaller than or equal to the terms of the
convergent p-series .

What can we conclude about the series ?

By the Comparison Test, this series converges.

What can we conclude about the original series, , which doesn’t have the absolute value signs?

By the absolute Convergence Theorem, the original series, , also converges.

Does this agree with the diagram above?

Yes.

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