Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55

CAL2.123 Limit Comparison Test Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.123 Limit Comparison Test. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.123 Limit Comparison Test - Select the correct benchmark series and limit comparison conclusion. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55.

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.123 Limit Comparison Test skill assessments:

Problem 1Difficulty: Mastery

Let a_n = (sqrt(n+1)-sqrt(n))/n. For sum_{n=1}^infty a_n, which statement correctly applies the Limit Comparison Test?

ACompare with sum 1/n; lim a_n/(1/n) is positive and finite; since sum 1/n diverges (harmonic series), the given series diverges.

BCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 1/2; since sum 1/n^(3/2) converges (p = 3/2 > 1), the given series converges.

CCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 0, so the standard Limit Comparison Test gives no convergence conclusion.

DCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 1/2; the given series diverges.

Solution & Explanation:Rationalizing gives 1/[n(sqrt(n+1)+sqrt(n))], which behaves like 1/(2n^(3/2)). Thus a_n/(1/n^(3/2)) -> 1/2, a positive finite number. Because sum 1/n^(3/2) converges (p = 3/2 > 1), the Limit Comparison Test shows that the given series converges. A common error is to see a difference of close terms and assume it behaves like either term alone; rationalize or use the derivative-sized difference.

Problem 2Difficulty: Mastery

A student needs a benchmark for sum_{n=1}^infty 1/sqrt(n) - 1/sqrt(n+1). Which choice correctly completes the limit comparison?

ACompare with sum 1/n; lim a_n/(1/n) is positive and finite; since sum 1/n diverges (harmonic series), the given series diverges.

BCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 0, so the standard Limit Comparison Test gives no convergence conclusion.

CCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 1/2; the given series diverges.

DCompare with sum 1/n^(3/2); lim a_n/(1/n^(3/2)) = 1/2; since sum 1/n^(3/2) converges (p = 3/2 > 1), the given series converges.

Solution & Explanation:Rationalizing the difference gives 1/[sqrt(n)sqrt(n+1)(sqrt(n+1)+sqrt(n))], which behaves like 1/(2n^(3/2)). Thus a_n/(1/n^(3/2)) -> 1/2, a positive finite number. Because sum 1/n^(3/2) converges (p = 3/2 > 1), the Limit Comparison Test shows that the given series converges. A common error is to see a difference of close terms and assume it behaves like either term alone; rationalize or use the derivative-sized difference.

Problem 3Difficulty: Mastery

Use the Limit Comparison Test for sum_{n=1}^infty 1/n - 1/(n+1). Which benchmark and conclusion are valid?

ACompare with sum 1/n²; lim a_n/(1/n²) = 1; since sum 1/n² converges (p = 2 > 1), the given series converges.

BCompare with sum 1/n; lim a_n/(1/n) is positive and finite; since sum 1/n diverges (harmonic series), the given series diverges.

CCompare with sum 1/n²; lim a_n/(1/n²) = 0, so the standard Limit Comparison Test gives no convergence conclusion.

DCompare with sum 1/n²; lim a_n/(1/n²) = 1; the given series diverges.

Solution & Explanation:The difference equals 1/[n(n+1)], which behaves like 1/n². Thus a_n/(1/n²) -> 1, a positive finite number. Because sum 1/n² converges (p = 2 > 1), the Limit Comparison Test shows that the given series converges. A common error is to see a difference of close terms and assume it behaves like either term alone; rationalize or use the derivative-sized difference.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55 (Select the correct benchmark series and limit comparison conclusion. Standards: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55.).