Unit 10 Practice Test

Infinite Sequences and Series

← Back to Unit 10

⏱️ Practice Test Instructions

⏰ Time Remaining: 45:00
📝 Instructions: This practice test contains 15 multiple-choice questions and 6 free-response questions. You have 45 minutes to complete the test. Use the timer to simulate real exam conditions. Check your answers as you go, and calculate your final score at the end.

📝 Multiple Choice Questions (15 questions)

1
Determine if the sequence aₙ = (n² + 1)/(2n² - 3) converges
2
Test ∑n=1∞ 1/(n² + 1) for convergence
3
Find the radius of convergence for ∑n=0∞ xⁿ/n!
4
Find the sum of ∑n=0∞ (1/3)ⁿ
5
Test ∑n=1∞ n!/nⁿ for convergence
6
Find the 3rd-degree Taylor polynomial for f(x) = ln(x) centered at x = 1
Note: Questions 7-15 follow the same format. Complete all multiple choice questions before proceeding to free response.

✏️ Free Response Questions (6 questions)

1
Test ∑n=1∞ n²/(n³ + 1) for convergence using the appropriate test

Solution:

1) Compare to the series ∑n=1∞ 1/n (harmonic series)

2) limn→∞ [n²/(n³ + 1)] / [1/n] = limn→∞ n³/(n³ + 1) = 1

3) Since the limit is a positive finite number and ∑1/n diverges

4) By the Limit Comparison Test, ∑n=1∞ n²/(n³ + 1) also diverges

Answer: The series diverges

2
Find the Maclaurin series for f(x) = eˣ and determine its radius of convergence

Solution:

1) f(x) = eˣ, so f⁽ⁿ⁾(x) = eˣ for all n

2) f⁽ⁿ⁾(0) = e⁰ = 1 for all n

3) Maclaurin series: eˣ = ∑n=0∞ (1/n!) xⁿ

4) = 1 + x + x²/2! + x³/3! + x⁴/4! + ...

5) For radius of convergence: R = limn→∞ |cₙ/cₙ₊₁| = limn→∞ (n+1) = ∞

6) Therefore, the radius of convergence is ∞ (converges for all x)

Answer: eˣ = ∑n=0∞ xⁿ/n!, R = ∞

3
Use the first three terms of the Maclaurin series for sin(x) to approximate sin(0.1)

Solution:

1) Maclaurin series for sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...

2) First three terms: sin(x) ≈ x - x³/6 + x⁵/120

3) For x = 0.1: sin(0.1) ≈ 0.1 - (0.1)³/6 + (0.1)⁵/120

4) = 0.1 - 0.001/6 + 0.00001/120

5) = 0.1 - 0.0001667 + 0.0000000833

6) ≈ 0.0998334

Answer: sin(0.1) ≈ 0.0998334

Note: Questions 4-6 follow the same format. Complete all free response questions before calculating your final score.

📊 Calculate Your Score

Your Performance

🎯 Study Recommendations: Congratulations on completing the final unit of AP Calculus BC! Review any topics you missed and retake this practice test. You've now mastered all the major concepts of AP Calculus BC.