Integral Test for Convergence

What the Integral Test Is For

The Integral Test gives you a way to determine whether an infinite series converges by connecting the series to an improper integral. Instead of trying to add infinitely many terms directly, you analyze the behavior of a related function as approaches infinity.

The basic idea is simple: if the corresponding improper integral has a finite value, the series converges. If the integral grows without bound, the series diverges.

This test is particularly useful for series involving powers of , logarithms, and other expressions that can be integrated relatively easily.

The Conditions You Must Check

The Integral Test does not apply automatically to every series. Suppose you have a series

You can use the Integral Test if the function corresponding to the terms is:

Positive: for all sufficiently large .
Continuous: is continuous on the interval being considered.
Decreasing: is decreasing for all sufficiently large .

When these conditions are satisfied, the series and its corresponding improper integral either both converge or both diverge.

The Integral Test Formula

If is positive, continuous, and decreasing for , then:

The important point is that you do not need to evaluate the integral to find the exact sum of the series. The Integral Test is a convergence test. You only need to determine whether the improper integral converges or diverges.

A Step-by-Step Method

Write the series as a function. Replace with so that .
Check the conditions. Verify that the function is positive, continuous, and decreasing for sufficiently large .
Set up the improper integral. Replace the infinite series with .
Evaluate the improper integral or its limit. Use and let .
State the conclusion. A finite limit means the series converges; an infinite limit means the series diverges.

Worked Examples

Example 1: A p-Series with p > 1

Determine whether the series converges or diverges.

Step 1: Define the corresponding function: .

Step 2: The function is positive and continuous for . It is also decreasing because .

Step 3: Apply the Integral Test:

Step 4: Evaluate the improper integral:

Result: The integral has a finite value, so the series converges.

Example 2: A Divergent p-Series

Determine whether the series converges or diverges.

Step 1: Define .

Step 2: The function is positive, continuous, and decreasing for .

Step 3: Apply the Integral Test:

Step 4: Evaluate the limit:

Result: The integral diverges, so the series diverges.

Example 3: A Series Involving a Logarithm

Determine whether the series converges or diverges.

Step 1: Let .

Step 2: For sufficiently large , the function is positive, continuous, and decreasing.

Step 3: Set up the improper integral:

Step 4: Use the substitution , giving .

Result: The improper integral diverges, so the series diverges.

Example 4: A Series with a Shifted Denominator

Determine whether the series converges or diverges.

Step 1: Define .

Step 2: The function is positive and continuous for . Its derivative is , which is negative for , so the function is decreasing.

Step 3: Apply the Integral Test:

Step 4: Evaluate:

Result: The integral approaches a finite value, so the series converges.

Why the Integral Test Works

The Integral Test works because the terms of a positive decreasing series can be compared to rectangles under the graph of its corresponding function.

For a decreasing function, the rectangles formed by the series terms provide upper and lower bounds for the area under the curve. Therefore, if the total area under the curve is finite, the total of the series terms must also remain finite. If the area becomes infinite, the series cannot converge.

You do not need to reproduce this geometric argument on most AP exam questions. What matters is recognizing when the conditions are satisfied and then carrying out the improper integral correctly.

Integral Test vs. the p-Series Test

The Integral Test is especially important because it provides the reasoning behind the familiar p-Series Test. Consider:

Applying the Integral Test gives:

Therefore:

So whenever you see a simple p-series, the p-Series Test is usually faster. But understanding the Integral Test explains why that result is true and helps you recognize related series that are not exact p-series.

Common Mistakes

Forgetting to check the conditions

You cannot simply write down an integral and declare that the Integral Test applies. The corresponding function must be positive, continuous, and decreasing for sufficiently large .

Thinking the integral gives the exact sum of the series

The Integral Test tells you whether the series converges or diverges. In general, the value of the improper integral is not the sum of the series.

Forgetting that the integral is improper

For an infinite series, the corresponding integral extends to infinity. You must write it as a limit such as before evaluating it.

Using the Integral Test when another test is much faster

If a series is clearly a p-series, use the p-Series Test. If the terms do not approach zero, use the nth Term Test for Divergence. The Integral Test is most useful when the integral is manageable and the series does not have an easier obvious test.

What the Integral Test Can and Cannot Tell You

If the Integral Test shows that a series converges, you know the infinite sum exists. If it shows that the integral diverges, the series diverges as well.

However, the Integral Test does not usually give you the exact value of the series. It also does not tell you how quickly the series converges by itself, although the associated integral can be used to develop remainder estimates for the error after a finite number of terms.

This makes the Integral Test useful not only for deciding convergence, but also for estimating how much of a convergent series you need to calculate to achieve a desired accuracy.

Practice Problems

Question 1

Determine whether the series converges or diverges using the Integral Test.

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Question 2

Determine whether the series converges or diverges using the Integral Test.

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Question 3

Determine whether the series converges or diverges.

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Question 4 (Application)

Use the Integral Test to determine whether the series converges or diverges.

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