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

CAL2.125 Alternating Series Remainder Bound Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.125 Alternating Series Remainder Bound. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.125 Alternating Series Remainder Bound - Select the correct error bound for an alternating-series approximation. 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 CAL2.125 Alternating Series Remainder Bound skill assessments:

Problem 1Difficulty: Mastery
For \(5\arctan(1/2)=5\sum_{n=0}\infty(-1)n\frac{(1/2)2n+1}{2n+1}\), terms \(n=0\) through \(n=3\) are used. What is the error bound for \(5\arctan(1/2)\)?
A\(\frac{5}{4608}\)
B\(\frac{1}{4608}\)
C\(\frac{5}{896}\)
D\(\frac{1}{512}\)
Solution & Explanation:For \(\arctan(1/2)\), the first omitted term after \(n=3\) is \((1/2)9/9=1/4608\). Multiplying the entire series by 5 gives error bound \(5/4608\).
Problem 2Difficulty: Mastery
For \(\sum_{n=1}\infty(-1)n+1\frac{1}{n2+n+1}\), what is the least \(N\) so the first \(N\) terms guarantee error at most \(0.0001\)?
A98
B100
C101
D99
Solution & Explanation:The bound is \(1/[(N+1)2+(N+1)+1]\). Testing \(N=98\) gives denominator \(9901<10000\), while \(N=99\) gives \(10101\), so the least \(N\) is 99.
Problem 3Difficulty: Mastery
Consider \(\sum_{k=1}\infty(-1)k\frac{k!}{10k}\). Which statement about using the alternating-series remainder bound is correct?
A\(∣R_N∣\le\frac{(N+1)!}{10N+1}\) for all \(N\)
B\(∣R_N∣\le\frac{N!}{10N}\)
CThe bound is not valid because the term magnitudes do not tend to 0
DThe error is exactly the first omitted term
Solution & Explanation:The magnitudes \(k!/10k\) eventually grow and do not tend to 0. Since the alternating-series hypotheses fail, the first omitted term is not a valid theorem bound.

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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 error bound for an alternating-series approximation. 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.).

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