Programmed tutorial: Comparison Tests of Convergence or Divergence of a Series: Example 1

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Determine whether the series
converges by using comparison tests.

General Contents

Detailed Contents

Index

First, because our various tests involve series starting with n = 1, split this series accordingly.

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 it appear that the series is converging?

Yes, the partial sums appear to be approaching approximately 0.22.

First, let’s use the general Comparison Test. State this test informally.

A series that is term by term “smaller” than a convergent series must also converge.
A series that is term by term “larger” than a divergent series must also diverge.

What is a convenient type of series to compare with the given series?

If we use

for comparison, we can determine it’s convergence or divergence via the Root Test.

Set up that Root Test.

Take the nth root.

We get

What can we conclude about this comparison series?

Since L < 1, the series converges.

How can we use this in the Comparison Test?

We need to determine if the terms

are smaller than the terms of

, the convergent series.

Are they?

Yes, because their denominators are larger than those of the other series.

What can we conclude now?

Since the terms of the series we are interested in,

, are smaller than the terms of the convergent series,

, we can conclude from the Comparison Test that both series converge.

Next, let’s try the Limit Comparison Test on this same series. Describe this test.

We compare two series of positive terms by taking the limit of their nth terms. If the limit is positive, then the two series either both converge or both diverge.

Set this up for

and

Simplify the division.

Do the division and take the limit.

What can we conclude from this comparison?

Since L2 > 0, either both series converge or both diverge. From our analysis of the comparison series as converging above, we conclude that both converge. This agrees with the conclusion above.

Does this agree with our diagram above?

Yes, we agree with the apparent convergence in the diagram.

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