General Contents

Detailed Contents

Programmed tutorial: Integral Test of Convergence or Divergence of a Series: 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.

Test the convergence of the series
with the Integral Test.

General Contents

Detailed Contents

First, plot the terms of this series on paper. To check, click “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 0.6.

Now let’s consider using the integral test to check on convergence. How do we decide whether this test is applicable?

We consider the function f(x), where f(n) = a_n.

What are the conditions on f(x)?

It must be continuous, positive, and decreasing
for x ≥ 1.

Is that true here?

Yes.

Write the f(x) which can be used here.

What integral do we use for the integral test?

We use

Substitute our f(x).

The integral becomes

Find the antiderivative.

Substitute the limits.

Simplify arctan(∞).

What can we conclude?

Since the integral converges, the series converges.

What can we find about the remainder, R_N?

We can find bounds for R_N.

How do we do this?

The remainder, R_N, is bounded by

Determine the bound on R₅ for this problem.

Find the decimal approximation to this bound.

Let’s use this bound to get an interval for S, the sum of the series.

To do so, we need S₅, the sum of the first 5 terms of this series. Determine S₅.

Combine these results to state the bounds on S.

We have found that R₅ could be 0, or as large as 0.190. Consequently
0.486 < S < 0.486 + 0.190, or
0.486 < S < 0.676.

Does this agree with our diagrams?

Yes, S appeared to approach a value in this range.

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