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Detailed Contents

Programmed tutorial: Ratio Test: Example 1

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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.

Determine whether the series
converges or diverges.

General Contents

Detailed Contents

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”.

alt="Java applet graph of the partial sums of the series for the ratio test." Your browser is completely ignoring the <APPLET> tag!

Does it appear that the series is converging?

Yes, the partial sums appear to be approaching approximately -1.

What convergence test is usually helpful when a factorial is present?

The Ratio Test.

Write the quantity we need to test for the Ratio Test.

We test

What condition on this limit indicates convergence of the series?

For convergence, Condition for convergence via the ratio test.

What condition on this limit indicates divergence?

For divergence, Condition for divergence via the ratio test.

Set up the limit for this series.

We need

“Invert and multiply”.

Find the absolute value of the (-1) dependence.

This takes care of all of the negative factors, so now we can look at just the remaining ratios.

How can we simplify the factor in L?

Upon cancelling 3ⁿ this becomes

How can we simplify the

factor in L?

We can use the fact that

Apply this fact here.

We get

.

Can we cancel the n!

Yes.

Do that.

This becomes

Combine these results to simplify the limit L.

,
because the denominator grows without bound.

Which part of the Ratio Test does this satisfy?

It satisfies

What can we conclude about the convergence or divergence?

The original series converges absolutely.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor