Integral Test for Convergence
Verify a positive, continuous, decreasing tail before using an improper integral to decide convergence.
College Board Topic 10.4 · CED p. 184 (PDF p. 189) · CED effective Fall 2020; project course outline 2026–27.
Suppose , and f is positive, continuous and decreasing for all sufficiently large x. Then and the improper integral either both converge or both diverge. A finite starting segment cannot change convergence, so it is enough for these conditions to hold on a tail.
Check the conditions before integrating. Positivity matters because the geometric comparison is between positive areas and positive terms. Continuity makes the improper integral legitimate on the tail; decreasing behavior supports the rectangle comparison. A function that merely passes through the data points is not automatically suitable.
The test compares convergence, not exact values. In general the sum does not equal the improper integral. The left and right rectangle pictures also give useful tail estimates: for positive decreasing f, . These bounds say how much lies beyond a chosen partial sum.
For beginning at n=2, choose . It is positive and continuous for x≥2 and decreasing there. With u=ln x, the improper integral is , a finite number. The series converges; its sum is not claimed to be 1/ln2.
Let's work through it
- Check f(x)=1/(x(ln x)²) on x≥2: it is positive, continuous, and . Why: The integral test needs a positive decreasing continuous tail.
- Use u=ln x to get . Why: The substitution has du=dx/x, and the reciprocal antiderivative tends to zero.
- The series converges. Why: Its associated improper integral is finite and the hypotheses hold.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Worked answer and reason
They justify the positive rectangle comparison.
Retrieval 2
Worked answer and reason
The test establishes matching convergence behavior, not equality of values.
Catch the tempting mistake
- Integration alone proves the test applies.
- First match f(n) to the terms and verify a positive, continuous, decreasing tail.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 10 problem 7 · No calculator · analytic
- Unit 10 problem 8 · No calculator · analytic
For the teacher
- Ask for a derivative or clear monotonic argument. Use a two-column sum/integral comparison to prevent an accidental equality claim.
Glossary and deeper reading · 5 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Improper integral
An improper integral has an infinite integration interval or an integrand that becomes unbounded within the interval or at an endpoint. It is defined using limits of proper integrals rather than substituting infinity as a number. For example, the integral of 1/x² from 1 to infinity converges to 1. If an interior singularity splits an interval, the separate improper integrals must both converge.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Partial sum
A partial sum adds the first finitely many terms of a series. The nth partial sum is Sₙ=a₁+a₂+⋯+aₙ when indexing begins at 1. For example, the first three terms of 1/2+1/4+1/8+⋯ give partial sum 7/8. Convergence of a series concerns the limit of these sums, not just the limit of the individual terms.
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Infinite series
An infinite series is the sum of the terms of an infinite sequence, defined through its partial sums. The series converges when those partial sums approach a finite value. For example, 1/2+1/4+1/8+⋯ converges to 1. Terms approaching 0 are necessary for convergence, but that condition alone is insufficient.