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

CAL2.130 Convergence of Series with Factorials Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.130 Convergence of Series with Factorials. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.130 Convergence of Series with Factorials - Choose the correct test and conclusion for a series containing factorial expressions. Standards tags: 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 CAL2.130 Convergence of Series with Factorials skill assessments:

Problem 1Difficulty: Mastery
Consider the series \(\sum_{n=1}\infty\frac{(2n)!}{3n n!(n+4)!}\). Which test and conclusion are correct?
ARatio Test: \(L=3/4\); the series converges.
BRatio Test: \(L=0\); the series converges absolutely.
CComparison with \(1/n4\) proves convergence.
DRatio Test: \(L=\frac43\); the series diverges.
Solution & Explanation:The unscaled factorial ratio tends to 4, and division by \(3n\) gives a Ratio Test limit of \(4/3\). Since \(4/3>1\), the series diverges.
Problem 2Difficulty: Mastery
Consider the series \(\sum_{n=1}\infty\frac{(2n)!}{5n n!(n+4)!}\). Which test and conclusion are correct?
ARatio Test: \(L=5/4\); the series diverges.
BRatio Test: \(L=1\); the test proves convergence.
CRatio Test: \(L=\frac45\); the series converges.
DThe nth-Term Test proves divergence because \((2n)!\) is in the numerator.
Solution & Explanation:For \((2n)!/(n!(n+4)!)\), the ratio tends to 4. The extra factor \(1/5n\) divides the limiting ratio by 5, so \(L=4/5<1\). The series converges.
Problem 3Difficulty: Mastery
Consider the series \(\sum_{n=1}\infty\frac{5n n!(n+4)!}{(2n)!}\). Which test and conclusion are correct?
ARatio Test: \(L=4/5\); the series converges.
BRatio Test: \(L=\frac54\); the series diverges.
CRatio Test: \(L=1\); the test proves convergence.
DComparison with \(1/n2\) proves convergence.
Solution & Explanation:The factorial part contributes a limiting ratio of \(1/4\), and \(5n\) multiplies it by 5, giving \(L=5/4\). Since this is greater than 1, the Ratio Test gives 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 (Choose the correct test and conclusion for a series containing factorial expressions. Standards tags: 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.).

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