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

CAL2.133 Communicating Series Conclusions Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.133 Communicating Series Conclusions. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.133 Communicating Series Conclusions - Select the precise written conclusion naming the test, conditions, and result. 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 CAL2.133 Communicating Series Conclusions skill assessments:

Problem 1Difficulty: Mastery

Which conclusion is precise for \(\sum_{n=1}^\inftyn²\bigl(\sqrt{n²+1}-n\bigr)\)?

AIt converges by comparison with \(\sum1/n²\).

BIt diverges by the nth-term divergence test because \(n²(\sqrt{n²+1}-n)=\frac{n²}{\sqrt{n²+1}+n}\sim\frac n2\), so the terms do not approach 0.

CIt converges by the Ratio Test because the square root expression approaches 0.

DIt converges conditionally because the terms are eventually decreasing.

Solution & Explanation:Although \(\sqrt{n²+1}-n\) is small, multiplying by \(n²\) makes the term grow like \(n/2\). The nth-term test proves divergence.

Problem 2Difficulty: Mastery

Which conclusion is precise for \(\sum_{n=1}^\infty\frac{\sqrt{n²+1}-n}{n}\)?

AIt converges by limit comparison with \(\sum1/n²\), since \(\frac{\sqrt{n²+1}-n}{n}=\frac{1}{n(\sqrt{n²+1}+n)}\sim\frac1{2n²}\).

BIt diverges by the nth-term test because \(\sqrt{n²+1}-n\to1\).

CIt diverges by comparison with \(\sum1/n\) because the expression contains a division by \(n\).

DIt converges conditionally by the Alternating Series Test.

Solution & Explanation:Rationalization gives behavior like \(1/(2n²)\), and a limit comparison with a convergent p-series proves convergence.

Problem 3Difficulty: Mastery

Which conclusion is precise for \(\sum_{n=3}^\infty\frac1{n\ln n\sqrt{\ln(\ln n)}}\)?

AIt converges by the Integral Test because the denominator has an extra square root factor.

BIt diverges by the Integral Test because after \(u=\ln(\ln x)\), the tail integral is \(\int u^-1/2\,du\), which diverges.

CIt converges by the p-series test with \(p=1/2\).

DThe Ratio Test proves divergence because the ratio limit is greater than 1.

Solution & Explanation:The integral-test substitution produces a p-integral in \(u\) with exponent \(1/2\le1\), so the integral and series diverge. This is not a direct p-series in \(n\).

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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 precise written conclusion naming the test, conditions, and result. 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.).