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

CAL2.119 Nth Term Test for Divergence Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.119 Nth Term Test for Divergence. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.119 Nth-Term Test for Divergence - Select whether the nth-term test proves divergence or is inconclusive. 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 CAL2.119 Nth Term Test for Divergence skill assessments:

Problem 1Difficulty: Mastery

A Calculus II student checks the terms of \(\sum_{n=2}^\infty \frac{\ln n}{\sqrt n}\). What should the nth-term test conclusion be?

AThe series must diverge because the formula contains \(n\).

BThe series converges by the nth-term test because \(a_n\to0\).

CThe series diverges by the nth-term test because \(a_n\to0\).

DInconclusive under the nth-term test, since the term limit is 0.

Solution & Explanation:Logarithmic growth is slower than any positive power of \(n\), so \(\ln n/\sqrt n\to0\). Because the term limit is 0, the nth-term test is inconclusive; it does not prove that the series converges. A common mistake is to treat a zero term limit as proof of convergence, but the nth-term test can only prove divergence.

Problem 2Difficulty: Mastery

Let \(a_n=\frac{n²}{n²+\ln n}\) be the nth term of an infinite series. What conclusion does the nth-term test give for \(\sum_{n=2}^\infty a_n\)?

ANo conclusion is possible from a nonzero term limit.

BThe series converges because the terms do not grow without bound.

CThe nth-term test proves divergence because the term limit is \(1\), which is nonzero.

DThe nth-term test is inconclusive because the term limit is finite.

Solution & Explanation:Dividing by \(n²\) gives \(1/(1+\ln n/n²)\to1\). Because the term limit is \(1\), not 0, the series diverges by the nth-term test. A common mistake is to treat a zero term limit as proof of convergence, but the nth-term test can only prove divergence.

Problem 3Difficulty: Mastery

Consider \(\sum_{n=1}^\infty \left(\frac{n²+1}{n²+n}\right)ⁿ\). Based on the limit of its terms, which statement is correct?

AThe nth-term test proves convergence whenever the terms have a limit.

BDiverges; the terms approach \(e^-1\), not 0.

CNo conclusion is possible from a nonzero term limit.

DThe series converges because the terms do not grow without bound.

Solution & Explanation:Taking logs gives \(n[\ln(1+1/n²)-\ln(1+1/n)]\to-1\), so the term limit is \(e^-1\). Because the term limit is \(e^-1\), not 0, the series diverges by the nth-term test. A common mistake is to treat a zero term limit as proof of convergence, but the nth-term test can only prove divergence.

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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 whether the nth-term test proves divergence or is inconclusive. 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).