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

CAL2.129 Choosing Between Ratio and Root Tests Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.129 Choosing Between Ratio and Root Tests. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.128 Root Test - Choose the convergence conclusion using the nth root of absolute terms. 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

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Below are sample questions representative of the exercises included on this worksheet and related CAL2.129 Choosing Between Ratio and Root Tests skill assessments:

Problem 1Difficulty: Mastery

Use the Root Test on ∑_{n=1}^∞ ((x²+4)/5)ⁿ/n. Which statement is correct?

AIt converges absolutely for -1 < x < 1, diverges for x < -1 or x > 1, and the Root Test is inconclusive at x = -1 and x = 1.

BIt converges absolutely for all real x because of the factor 1/n.

CIt diverges for all real x because x²+4 is positive.

DThe Root Test is inconclusive for all real x because n^(1/n) tends to 1.

Solution & Explanation:The nth root of the absolute term is ((x²+4)/5)/n^(1/n), which tends to (x²+4)/5. This is less than 1 when x² < 1, greater than 1 when x² > 1, and equal to 1 at x = ±1. The factor 1/n does not affect the nth-root limit.

Problem 2Difficulty: Mastery

Use the Root Test on ∑_{n=1}^∞ (x²/(1+x²))ⁿ for real x. Which statement is correct?

AThe series diverges for every real x because x² is nonnegative.

BThe Root Test is inconclusive for every real x.

CThe series converges only when x = 0.

DThe series converges absolutely for every real x by the Root Test.

Solution & Explanation:The nth root of the absolute term is x²/(1+x²). For every real x, this value is less than 1, with value 0 at x = 0. Therefore the Root Test proves absolute convergence for every real x. Nonnegativity of the terms is not the deciding condition; the root limit being below 1 is.

Problem 3Difficulty: Mastery

Use the Root Test on ∑_{n=1}^∞ (x-1)^(2n)/3ⁿ. Which statement is correct?

AIt converges absolutely only when x = 1.

BIt diverges whenever x is negative.

CIt converges absolutely when (x-1)² < 3, diverges when (x-1)² > 3, and the Root Test is inconclusive when (x-1)² = 3.

DIt is inconclusive for all x because the exponent is 2n.

Solution & Explanation:The nth root of the absolute term is (x-1)²/3. Compare (x-1)²/3 with 1. The even power removes sign issues, but it does not make every value of x converge.

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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 (Choose the convergence conclusion using the nth root of absolute terms. 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.).