Integral test for convergence — AP Calculus BC Study Guide

For: AP Calculus BC candidates sitting AP Calculus BC.

Covers: The three hypotheses of the integral test for convergence, step-by-step application to positive-term infinite series, remainder error bounds for partial sums, and connections to improper integral evaluation for AP Calculus BC.

You should already know: How to evaluate improper integrals of all types. Properties of positive-term infinite series. How to compute derivatives and antiderivatives of basic functions.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus BC style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Integral test for convergence?

The integral test for convergence is a core convergence test for infinite series that relates the behavior of an infinite discrete sum to the area under a continuous function, leveraging existing knowledge of improper integrals. It is exclusively used for positive-term series, and it is a key topic in Unit 10 (Infinite Sequences and Series) of the AP Calculus BC Course and Exam Description (CED). Unit 10 accounts for 17–18% of the total AP exam score, and the integral test appears in both multiple-choice (MCQ) and free-response (FRQ) sections. MCQs often ask to verify test conditions or identify a convergent series, while FRQs may pair the integral test with error bound questions or connect it to other convergence tests. The key idea is that for a series ${\sum }_{n=1}^{\infty }{a}_n$ where $a_n=f(n)$, the convergence behavior of the series matches the convergence behavior of the improper integral ${\int }_1^{\infty }f(x)dx$. It is sometimes called the Cauchy-Maclaurin integral test, but AP exclusively refers to it as the integral test for convergence.

2. Hypotheses of the Integral Test

The integral test cannot be applied to just any series: it requires three strict conditions (hypotheses) to hold to produce a valid conclusion. These conditions are non-negotiable on the AP exam—failing to verify them will cost you points on FRQs. The three hypotheses, for a function $f(x)$ defined such that $a_n=f(n)$ for all integers $n\ge N$ (usually $N=1$, but can be any positive integer), are:

1. $f$ is continuous on $[N,\infty )$: no vertical asymptotes, jumps, or discontinuities in the interval starting at $N$.
2. $f$ is positive on $[N,\infty )$: every output of $f$ for $x\ge N$ is greater than 0.
3. $f$ is decreasing on $[N,\infty )$: as $x$ increases, $f(x)$ does not increase, which is equivalent to $f^{\prime }(x)\le 0$ for all $x>N$.

These conditions matter because the test compares the series sum to Riemann sums of the integral: a decreasing positive function ensures the series is always bounded between two consecutive integrals, so convergence of one implies convergence of the other. If any condition is violated, the test gives no valid conclusion.

Worked Example

Problem: Does the integral test apply to the series ${\sum }_{n=2}^{\infty }\frac{\ln n}{n}$? Verify all hypotheses to justify your answer.

1. Define $f(x)=\frac{\ln x}{x}$ for $x\ge 2$. Check continuity: $f(x)$ is a quotient of two continuous functions, and the denominator $x$ is non-zero for $x\ge 2$, so $f$ is continuous on $[2,\infty )$.
2. Check positivity: For $x\ge 2$, $\ln x>\ln 2>0$ and $x>0$, so $f(x)>0$ on $[2,\infty )$.
3. Check if $f$ is decreasing: Compute the derivative: $f^{\prime }(x)=\frac{(\frac{1}{x})x-\ln x(1)}{x^2}=\frac{1-\ln x}{x^2}$.
4. For $x>e\approx 2.718$, $\ln x>1$, so $1-\ln x<0$, meaning $f^{\prime }(x)<0$ on $(e,\infty )$. We can shift the starting index to $n=3$ (since convergence only depends on the tail of the series), so all hypotheses are satisfied starting at $N=3$. Conclusion: Yes, the integral test can be applied to this series.

Exam tip: On AP FRQs, if you are asked to use the integral test, you must explicitly verify all three hypotheses to earn full credit—never skip this step, even if the conditions seem obvious.

3. Applying the Integral Test to Determine Convergence

Once you have verified all three hypotheses, the integral test rule is straightforward: the infinite series ${\sum }_{n=N}^{\infty }{a}_{n}convergesifandonlyiftheimproperintegral$\int_N^\infty f(x) dx$ converges. If the integral evaluates to a finite number, the series converges; if the integral diverges (grows without bound), the series also diverges.

This relationship comes from the Riemann sum bound: for a decreasing positive function, $${\int }_N^{\infty }f(x)dx\le \sum _{n=N}^{\infty }{a}_n\le a_N+{\int }_N^{\infty }f(x)dx$$ This means the series is bounded between two finite values if the integral is finite, and grows without bound if the integral does. The most famous result from the integral test is the convergence rule for p-series: ${\sum }_{n=1}^{\infty }\frac{1}{n^p}$ converges if $p>1$ and diverges if $p\le 1$, which is proven directly with the integral test.

Worked Example

Problem: Determine whether the series ${\sum }_{n=1}^{\infty }\frac{1}{n^2+1}$ converges or diverges using the integral test.

1. Define $f(x)=\frac{1}{x^2+1}$ for $x\ge 1$. Verify hypotheses: (a) Continuous: denominator is never zero for all real $x$, so $f$ is continuous on $[1,\infty )$. (b) Positive: $x^2+1>0$ for all $x$, so $f(x)>0$. (c) Decreasing: $f^{\prime }(x)=\frac{-2x}{(x^2+1{)}^2}<0$ for all $x>0$, so $f$ is decreasing. All hypotheses are satisfied.
2. Evaluate the improper integral: $${\int }_1^{\infty }\frac{1}{x^2+1}dx=\underset{b\to \infty }{\lim }{\int }_1^b\frac{1}{x^2+1}dx=\underset{b\to \infty }{\lim }{\left[\arctan x\right]}_1^b$$
3. Compute the limit: ${\lim }_{b\to \infty }(\arctan b-\arctan 1)=\frac{\pi }{2}-\frac{\pi }{4}=\frac{\pi }{4}$, which is finite.
4. By the integral test, since the integral converges, the series converges.

Exam tip: Remember that the integral test only tells you if the series converges—it never tells you what the sum of the series is. A common MCQ distractor is an option equal to the value of the integral, which is almost never the sum of the series.

4. Remainder Error Bounds for Integral Test

When you approximate the sum of a convergent infinite series with its $n$-th partial sum $S_n$, the remainder $R_n=S-S_n$ is the error in your approximation. For series that satisfy the integral test hypotheses, we can find explicit upper and lower bounds for this error using the same improper integral from the convergence test.

Specifically, if $f$ is continuous, positive, and decreasing for $x\ge n$, and ${\sum }_{k=1}^{\infty }f(k)=S$ converges, then the remainder satisfies: $${\int }_{n+1}^{\infty }f(x)dx\le R_n\le {\int }_n^{\infty }f(x)dx$$ This bound comes directly from the Riemann sum comparison: the right Riemann sum gives a lower bound for the error, and the left Riemann sum gives an upper bound. Explicit error bounds are regularly tested on AP FRQs, so this is a high-yield topic to master.

Worked Example

Problem: The series ${\sum }_{n=1}^{\infty }\frac{1}{n^3}$ is known to converge. Find an upper and lower bound for the error when the sum is approximated by the 10th partial sum $S_{10}$.

1. Confirm $f(x)=\frac{1}{x^3}$ satisfies all integral test hypotheses for $x\ge 1$: continuous, positive, and $f^{\prime }(x)=-3x^{-4}<0$ so decreasing. The remainder bound formula applies.
2. Lower bound for $R_{10}$ is ${\int }_{11}^{\infty }\frac{1}{x^3}dx$. Evaluate: ${\lim }_{b\to \infty }{\int }_{11}^b{x}^{-3}dx={\lim }_{b\to \infty }{\left[-\frac{1}{2x^2}\right]}_{11}^b=0-(-\frac{1}{2(121)})=\frac{1}{242}\approx 0.00413$.
3. Upper bound for $R_{10}$ is ${\int }_{10}^{\infty }\frac{1}{x^3}dx=\frac{1}{2(10{)}^2}=0.005$.
4. Conclusion: The error $R_{10}$ satisfies $0.00413\le R_{10}\le 0.005$.

Exam tip: If a question only asks for an error bound (not both upper and lower), it will always accept the upper bound ${\int }_n^{\infty }f(x)dx$ as the correct answer.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Claiming the integral test applies to ${\sum }_{n=1}^{\infty }\frac{\sin n}{n^2}$ because ${\int }_1^{\infty }\frac{\sin x}{x^2}dx$ converges. Why: The student forgot to check the positivity condition, and $\frac{\sin x}{x^2}$ alternates sign, violating the second hypothesis. Correct move: Always check all three hypotheses (continuous, positive, decreasing) before applying the integral test, and never use the test on alternating or term-negative series.
• Wrong move: Concluding ${\sum }_{n=1}^{\infty }\frac{1}{n}$ converges because ${\int }_1^{1000}\frac{1}{x}dx=\ln 1000$ is finite. Why: Students confuse definite integrals with improper integrals, forgetting we need the integral to infinity to apply the test. Correct move: Always evaluate the improper integral by taking the limit as the upper bound goes to infinity, never stop at a finite bound.
• Wrong move: Claiming ${\sum }_{n=1}^{\infty }\frac{1}{n^2}$ converges to 1 because ${\int }_1^{\infty }\frac{1}{x^2}dx=1$. Why: Students confuse the value of the integral with the sum of the series, a common misconception from how the test is structured. Correct move: Remember that the integral test only determines convergence or divergence, it never gives the value of the series sum.
• Wrong move: Stating the series diverges via the integral test when ${\lim }_{n\to \infty }{a}_n\ne 0$. Why: Students mix up the nth term test with the integral test, and incorrectly attribute the divergence conclusion to the wrong test. Correct move: If ${\lim }_{n\to \infty }{a}_n\ne 0$, use the nth term test for divergence to conclude divergence, do not invoke the integral test for this conclusion.
• Wrong move: Using the remainder bound formula for a divergent series to bound partial sum error. Why: Students memorize the formula without remembering it only applies to convergent series, since divergent series have no finite sum to approximate. Correct move: Only use the integral test remainder bound after you have confirmed the series converges via the integral test.

6. Practice Questions (AP Calculus BC Style)

Question 1 (Multiple Choice)

For which of the following series can the integral test for convergence be correctly applied, starting at the lower bound of the series as given? A) ${\sum }_{n=1}^{\infty }\frac{(-1{)}^n}{n}$ B) ${\sum }_{n=1}^{\infty }\frac{e^{-n}}{n^2}$ C) ${\sum }_{n=1}^{\infty }\frac{\cos n}{n^3}$ D) ${\sum }_{n=1}^{\infty }\frac{1}{1-n^2}$

Worked Solution: To answer, we check the three required hypotheses (continuous, positive, decreasing) for each option. Option A has negative terms from alternating signs, violating the positivity requirement, so A is incorrect. Option C has terms that are positive for some $n$ and negative for others, so it also violates positivity, so C is incorrect. Option D has an undefined term at $n=1$ (division by zero), so the function is not continuous at $x=1$, violating the continuity requirement, so D is incorrect. For Option B, $f(x)=\frac{e^{-x}}{x^2}$ is continuous, positive, and decreasing (its derivative is negative for all $x>0$) on $[1,\infty )$, so all hypotheses are satisfied. Correct answer: $\boxed{B}$.

Question 2 (Free Response)

Consider the infinite series ${\sum }_{n=1}^{\infty }\frac{x}{n^2}$ for $x>0$. (a) Show that all three hypotheses of the integral test are satisfied for this series. (b) Use the integral test to determine for what values of $x$ the series converges. (c) Given $x=1$, find an upper bound for the error when the series is approximated by the 50th partial sum $S_{50}$.

Worked Solution: (a) Let $f(t)=\frac{x}{t^2}$ for $t\ge 1$ and $x>0$. (1) Continuity: $f(t)$ is a rational function with non-zero denominator for $t\ge 1$, so it is continuous. (2) Positivity: $x>0$ and $t^2>0$ for all $t\ge 1$, so $f(t)>0$. (3) Decreasing: $f^{\prime }(t)=-\frac{2x}{t^3}<0$ for all $t>0$, so $f$ is decreasing on $[1,\infty )$. All three hypotheses are satisfied. (b) Evaluate the improper integral: $${\int }_1^{\infty }\frac{x}{t^2}dt=x\underset{b\to \infty }{\lim }{\left[-\frac{1}{t}\right]}_1^b=x\underset{b\to \infty }{\lim }\left(1-\frac{1}{b}\right)=x$$ For any $x>0$, the integral converges to a finite value $x$, so by the integral test, the series converges for all $x>0$. (c) The upper bound for the error $R_{50}$ when $x=1$ is: $$R_{50}\le {\int }_{50}^{\infty }\frac{1}{t^2}dt=\frac{1}{50}=0.02$$ The upper bound for the error is $0.02$.

Question 3 (Application / Real-World Style)

A regional power grid operator models the annual additional generation capacity added by incremental grid upgrade projects as $c_n=\frac{150}{n^{1.1}}$ megawatts (MW), for $n=1,2,3,...$ where $n$ is the $n$-th upgrade. The total long-run additional capacity from all possible future upgrades is given by the infinite series ${\sum }_{n=1}^{\infty }{c}_n$. Use the integral test to answer the following: (1) Is the total long-run additional capacity finite? (2) If the operator approximates the total capacity using just the first 10 upgrades, what is the maximum possible error in this approximation?

Worked Solution:

1. Verify hypotheses for $f(x)=\frac{150}{x^{1.1}}$ on $x\ge 1$: $f$ is continuous, positive, and decreasing (the derivative $f^{\prime }(x)=-165x^{-2.1}<0$ for all $x>0$), so the integral test applies. Evaluate the improper integral: $${\int }_1^{\infty }\frac{150}{x^{1.1}}dx=150\underset{b\to \infty }{\lim }{\left[\frac{x^{-0.1}}{-0.1}\right]}_1^b=1500$$ The integral converges, so the series converges, meaning total long-run additional capacity is finite.
2. The maximum error (upper bound for $R_{10}$) is: $$R_{10}\le {\int }_{10}^{\infty }\frac{150}{x^{1.1}}dx=\frac{1500}{10^{0.1}}\approx 1192\text{ MW}$$ In context: The true total long-run capacity is no more than 1192 MW higher than the approximation from the first 10 upgrades.

7. Quick Reference Cheatsheet

8. What's Next

The integral test is the foundation for most other convergence tests for infinite series, and it is a prerequisite for the direct and limit comparison tests that come next in the AP Calculus BC syllabus. Understanding how the tail of a series behaves relative to an improper integral also builds intuition for all error bound questions, which appear regularly in FRQs on alternating series and power series approximation. Without mastering the hypotheses of the integral test, you will often incorrectly justify convergence tests and lose points on free-response questions. The integral test also reinforces the core AP Calculus connection between discrete sums and continuous integration. Next topics to study: Direct comparison test Limit comparison test Alternating series convergence and error bounds P-series convergence