Programmed tutorial: Alternating Series: Convergence or Divergence: Example 1

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Test whether the alternating series
converges.

General Contents

Detailed Contents

Index

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 -0.06.

How can we show that this series is convergent?

Since it is an alternating series, we can try to show that

for n ≥ 1,
and that

First, let’s use the general Comparison Test. Set it up.

Do both the numerator and the denominator each approach ∞?

Yes.

How do we proceed?

We need to apply L’Hopital’s Rule. Describe this rule.

We differentiate the numerator and the denominator separately, take the ratio, and evaluate the limit of the resulting expression. (We do not use the Quotient Rule).

We need a continuous function of x for L'Hopital's Rule. What function should we use?

We can use