The Integral Test is a method for deciding whether an infinite series converges or diverges by comparing it to an improper integral. It works when the terms of the series come from a function that is positive, continuous, and decreasing on a suitable interval. The test matters because many series are hard to add directly, but their related integrals are easier to evaluate or estimate.
Visually, the test connects the sum of infinitely many rectangle areas to the area under a curve.
If a_n = f(n), then the series sum from n = 1 to infinity of a_n is compared with the improper integral from 1 to infinity of f(x) dx. When the curve decreases, the rectangles representing f(n) can be placed above or below the curve to trap the series near the integral. The test says the series and the integral share the same convergence behavior, although they usually do not have the same value.
A major application is the p-series, where sum 1/n^p converges exactly when p > 1.
Understanding Calculus: The Integral Test
The important idea is that convergence is decided by the tail, meaning the terms far out in the series. Adding, removing, or changing a fixed number of early terms changes the total by only a finite amount. It cannot turn a convergent infinite sum into a divergent one.
This is why the test may begin at a convenient whole number rather than at one. In practice, a formula may fail to decrease near the start but behave properly after some point. Students should state where that point begins and then analyze only the remaining terms.
The rectangle picture gives more than a visual memory aid. For a decreasing curve, a rectangle built from the value at a left endpoint sits above the curve over its interval. A rectangle built from the value at a right endpoint sits below it.
These two sets of rectangles form upper and lower bounds for the area. Their difference is controlled by a few boundary terms, not by an unlimited new error. That fact explains why the sum and the improper integral can have different numerical values while still sharing the same finite or infinite behavior.
To use the test carefully, first create a function by replacing the index with a real variable. Then inspect its behavior far to the right. Positivity matters because negative areas could cancel positive ones and hide the size of the terms.
Continuity prevents gaps that would make the area comparison unreliable. Decreasing behavior is what makes the rectangle bounds point in the correct direction. Derivatives are often useful here.
If the derivative is negative after a certain value, the function decreases there. A function does not need to decrease forever from zero. It only needs suitable behavior on the tail being tested.
A common challenge comes from expressions involving logarithms. The series with terms one divided by n times the natural logarithm of n is a classic borderline case. Its related integral becomes the integral of one divided by x times the natural logarithm of x.
A substitution using the natural logarithm turns this into an integral of one divided by a new variable, which grows without bound. Therefore the series diverges, even though its terms become very small. This teaches an essential lesson.
Terms approaching zero are necessary for convergence, but they are not enough. When working examples, pay attention to the growth rate in the denominator and to whether an antiderivative approaches a finite limit as the upper bound increases.
Key Facts
- Integral Test conditions: f(x) must be positive, continuous, and decreasing for x >= N.
- If a_n = f(n), compare sum from n = N to infinity of a_n with integral from N to infinity of f(x) dx.
- If integral from N to infinity of f(x) dx converges, then sum from n = N to infinity of a_n converges.
- If integral from N to infinity of f(x) dx diverges, then sum from n = N to infinity of a_n diverges.
- Improper integral form: integral from N to infinity of f(x) dx = limit as b -> infinity of integral from N to b of f(x) dx.
- p-series result: sum from n = 1 to infinity of 1/n^p converges if p > 1 and diverges if p <= 1.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, written as a_1 + a_2 + a_3 + ... .
- Convergence
- Convergence means the sequence of partial sums approaches a finite number.
- Improper integral
- An improper integral is an integral with an infinite limit or an integrand that becomes unbounded.
- Decreasing function
- A decreasing function has values that do not increase as x moves to the right.
- p-series
- A p-series is a series of the form sum from n = 1 to infinity of 1/n^p.
Common Mistakes to Avoid
- Ignoring the conditions of the Integral Test: the function must be positive, continuous, and decreasing eventually, or the test may not apply.
- Thinking the series equals the integral: the Integral Test compares convergence behavior, but the sum and the integral usually have different numerical values.
- Starting at the wrong place: changing or removing finitely many beginning terms does not affect convergence, so the test only needs to work for all sufficiently large n.
- Misapplying the p-series rule: sum 1/n^p converges only when p > 1, so the harmonic series with p = 1 diverges.
Practice Questions
- 1 Use the Integral Test to determine whether sum from n = 1 to infinity of 1/n^2 converges or diverges.
- 2 Use the Integral Test to determine whether sum from n = 2 to infinity of 1/(n ln n) converges or diverges.
- 3 Explain why the Integral Test compares convergence but does not usually give the exact value of an infinite series.