Calculus II · Unit 4A · lesson

The Integral Test

Concept

Learning objectives

use an improper integral to determine convergence of a positive decreasing series.

The Integral Test

Explanation

Sums and areas can share the same tail behavior

When $f$ is positive, continuous, and decreasing, the values $f(n)$ can be represented by rectangle heights. The series adds rectangle areas of width one, while the improper integral measures area under the curve. Because the rectangles and curve trap one another up to a finite initial difference, either both tails are finite or both are infinite.

The hypotheses matter. Positivity prevents cancellation from disguising size. Decreasing behavior gives a consistent rectangle comparison. Continuity keeps the integral well behaved. A correct solution identifies a function $f$ , verifies the hypotheses on a tail interval, evaluates the improper integral, and then states the corresponding conclusion for the series.

Bridge

A series and an area can measure the same tail

For a positive decreasing function, the rectangles of height $f(n)$ and the area under $y=f(x)$ trap one another up to a finite initial difference. If the improper integral has finite area, the corresponding series has finite total; if the area is infinite, the series diverges.

The hypotheses are part of the theorem, not decorative paperwork. Positivity prevents cancellation from confusing size, continuity makes the integral comparison ordinary, and eventual decrease makes the rectangle inequalities point consistently. The behavior of finitely many early terms never affects convergence, so the conditions only need to hold eventually.

**Decreasing rectangles trap the integral tail**
A positive decreasing curve is shown with unit-width rectangles. Right-endpoint rectangles lie below the curve and left-endpoint rectangles lie above it.
Decreasing rectangles trap the integral tail. Decreasing curve with left and right rectangles bracketing the tail.

Proof idea

The rectangle inequalities differ only at the boundary

Decision

Integral Test workflow

Write $a_n=f(n)$ . Verify that $f$ is positive, continuous, and decreasing for all sufficiently large inputs. Evaluate the improper integral with an explicit limit. Finally, transfer only the convergence conclusion to the series; the series sum is not generally equal to the integral.

For decreasing $f$ ,

\int_1^{N+1} f(x)\,dx\le \sum_{n=1}^{N}f(n) \le f(1)+\int_1^N f(x)\,dx.

These inequalities force the series and improper integral to converge or diverge together.

**Series terms and areas under a decreasing curve**
For a positive decreasing function, the rectangles associated with $f(n)$ can be compared directly with the improper integral under $f(x)$ .
Series terms and areas under a decreasing curve. Decreasing curve with left and right rectangles bracketing the tail.

How to read the visual

Because $1/x$ decreases, each left-endpoint rectangle lies above the curve on its interval. Divergence of the area therefore forces divergence of the harmonic-series rectangles.

Concept

Integral Test

If $f$ is positive, continuous, and decreasing for $x\ge N$ , and $a_n=f(n)$ , then

\sum_{n=N}^{\infty}a_n \quad\text{and}\quad \int_N^{\infty}f(x)\,dx

either both converge or both diverge.

Guided walkthrough

A logarithmic example

Consider

\sum_{n=2}^{\infty}\frac1{n(\ln n)^2}.

With $f(x)=1/[x(\ln x)^2]$ , substitute $u=\ln x$ :

\int_2^{\infty}\frac{dx}{x(\ln x)^2} =\int_{\ln2}^{\infty}u^{-2}\,du =\frac1{\ln2}.

The integral converges, so the series converges.

Worked example

A logarithmic correction to the harmonic series

Test

\sum_{n=2}^{\infty}\frac{1}{n(\ln n)^2}.

Let $f(x)=1/[x(\ln x)^2]$ , which is positive, continuous, and decreasing for $x\ge2$ . Then

\int_2^\infty \frac{dx}{x(\ln x)^2} =\left[-\frac1{\ln x}\right]_2^\infty =\frac1{\ln2}.

The improper integral converges, so the series converges.

Common mistake

Do not forget to verify the hypotheses

Writing an improper integral beside a series is not yet an Integral Test argument. State positivity, continuity, and eventual decrease, then evaluate the integral.

Interactive checku4a-integral_test-01

Use the Integral Test to classify $\sum_{n=1}^{\infty}1/n^2$ .

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Compare with $\int_1^{\infty}x^{-2}\,dx$ .

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Exercise

Use the Integral Test on $\sum1/n$ .

Exercise

Classify $\sum_{n=3}^{\infty}1/[n\ln n]$ .

Exercise

Explain why changing finitely many initial terms does not affect convergence.

Exercise

Identify a positive sequence for which the displayed interpolation is not decreasing and explain how to begin farther out.

After the explanation

Use the section idea

Reading lens

Compare positive terms by long-run size and select a benchmark whose convergence behavior is already known.

Mental model

Direct comparison transfers inequalities; limit comparison transfers asymptotic scale; the integral test links sums to accumulated area.

Decision

Use a clean inequality when available, asymptotic comparison when ratios stabilize, and the integral test when a matching decreasing function is natural.

Common trap

Reversing the direction needed to prove convergence or divergence, or forgetting an integral-test remainder condition.

Check yourself

Does your benchmark support the conclusion in the direction you claim, and are all hypotheses stated?

Source & rights

Original instruction with traceable references.

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