Unit 10: Infinite Sequences and Series

Master the culmination of AP Calculus BC with convergence, power series, and Taylor polynomials

← Back to Units

📋 Unit Overview

🎯 Learning Objectives: This unit represents the pinnacle of AP Calculus BC, exploring infinite processes and their applications in approximating functions and solving complex problems.

Unit 10 introduces the fascinating world of infinite sequences and series, where we explore convergence, power series, and Taylor polynomials. These concepts are essential for understanding advanced mathematics and have applications in physics, engineering, and computer science.

Key Topics:
  • Infinite sequences and convergence
  • Infinite series and convergence tests
  • Power series and radius of convergence
  • Taylor and Maclaurin series
  • Function approximation
  • Error bounds and remainders
Skills You'll Master:
  • Determining sequence convergence
  • Applying convergence tests
  • Finding power series representations
  • Constructing Taylor polynomials
  • Estimating function values
  • Analyzing error in approximations

🔢 Infinite Sequences

Basic Concepts

A sequence is an ordered list of numbers: {a₁, a₂, a₃, ...} or {aₙ}

💡 Key Definitions:
  • Convergent: limn→∞ aₙ = L (finite limit)
  • Divergent: limn→∞ aₙ does not exist or is infinite
  • Monotonic: Always increasing or always decreasing
  • Bounded: All terms lie between two fixed numbers
Example 1: Sequence Convergence

Determine if the sequence aₙ = (n² + 1)/(2n² - 3) converges.

Solution:

1) limn→∞ (n² + 1)/(2n² - 3)

2) Divide by highest power: limn→∞ (1 + 1/n²)/(2 - 3/n²)

3) = (1 + 0)/(2 - 0) = 1/2

4) The sequence converges to 1/2.

➕ Infinite Series

Definition and Convergence

An infinite series is the sum of an infinite sequence:

∑n=1∞ aₙ = a₁ + a₂ + a₃ + ...

The series converges if the sequence of partial sums Sₙ = a₁ + a₂ + ... + aₙ converges.

Example 2: Geometric Series

Determine if ∑n=0∞ (1/3)ⁿ converges and find its sum.

Solution:

1) This is a geometric series with a = 1 and r = 1/3

2) Since |r| = 1/3 < 1, the series converges

3) Sum = a/(1-r) = 1/(1-1/3) = 1/(2/3) = 3/2

🔍 Convergence Tests

Essential Tests

Basic Tests:
  • nth Term Test: If lim aₙ ≠ 0, series diverges
  • Integral Test: Compare to ∫f(x)dx
  • p-Series: ∑1/nᵖ converges if p > 1
  • Geometric: ∑arⁿ converges if |r| < 1
Comparison Tests:
  • Direct Comparison: Compare term by term
  • Limit Comparison: Compare ratios
  • Ratio Test: lim |aₙ₊₁/aₙ|
  • Root Test: lim ⁿ√|aₙ|
Example 3: Ratio Test

Test ∑n=1∞ n!/nⁿ for convergence.

Solution:

1) aₙ = n!/nⁿ

2) aₙ₊₁ = (n+1)!/(n+1)ⁿ⁺¹

3) |aₙ₊₁/aₙ| = [(n+1)!/(n+1)ⁿ⁺¹] / [n!/nⁿ]

4) = (n+1) · nⁿ / (n+1)ⁿ⁺¹ = nⁿ / (n+1)ⁿ

5) = (n/(n+1))ⁿ = (1/(1+1/n))ⁿ → 1/e < 1

6) Since limit < 1, series converges by Ratio Test

⚡ Power Series

Definition and Radius of Convergence

A power series centered at x = a is:

∑n=0∞ cₙ(x-a)ⁿ = c₀ + c₁(x-a) + c₂(x-a)² + ...
🔍 Radius of Convergence:
  • Use Ratio Test: R = lim |cₙ/cₙ₊₁|
  • If R = 0: converges only at center
  • If R = ∞: converges for all x
  • If 0 < R < ∞: converges for |x-a| < R
Example 4: Finding Radius of Convergence

Find the radius of convergence for ∑n=0∞ xⁿ/n!

Solution:

1) cₙ = 1/n!

2) |cₙ₊₁/cₙ| = (1/(n+1)!) / (1/n!) = n!/(n+1)! = 1/(n+1)

3) R = limn→∞ |cₙ/cₙ₊₁| = limn→∞ (n+1) = ∞

4) Radius of convergence is ∞ (converges for all x)

📐 Taylor and Maclaurin Series

Taylor Series Formula

The Taylor series for f(x) centered at x = a is:

f(x) = ∑n=0∞ [f⁽ⁿ⁾(a)/n!] (x-a)ⁿ

When a = 0, this becomes the Maclaurin series.

Example 5: Maclaurin Series for eˣ

Find the Maclaurin series for f(x) = eˣ.

Solution:

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

2) f⁽ⁿ⁾(0) = e⁰ = 1 for all n

3) eˣ = ∑n=0∞ (1/n!) xⁿ

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

📚 Common Maclaurin Series:
  • eˣ = ∑n=0∞ xⁿ/n!
  • sin(x) = ∑n=0∞ (-1)ⁿ x²ⁿ⁺¹/(2n+1)!
  • cos(x) = ∑n=0∞ (-1)ⁿ x²ⁿ/(2n)!
  • 1/(1-x) = ∑n=0∞ xⁿ for |x| < 1

🎯 Function Approximation

Taylor Polynomials

The nth-degree Taylor polynomial is the first n+1 terms of the Taylor series:

Pₙ(x) = ∑k=0n [f⁽ᵏ⁾(a)/k!] (x-a)ᵏ
Example 6: Taylor Polynomial

Find the 3rd-degree Taylor polynomial for f(x) = ln(x) centered at x = 1.

Solution:

1) f(x) = ln(x), f(1) = 0

2) f'(x) = 1/x, f'(1) = 1

3) f''(x) = -1/x², f''(1) = -1

4) f'''(x) = 2/x³, f'''(1) = 2

5) P₃(x) = 0 + 1(x-1) + (-1/2)(x-1)² + (2/6)(x-1)³

6) = (x-1) - (x-1)²/2 + (x-1)³/3

🎯 Practice Problems

Problem 1: Sequence Convergence

Determine if aₙ = (3n² + 2n)/(n² + 1) converges.

Solution:

1) limn→∞ (3n² + 2n)/(n² + 1)

2) Divide by n²: limn→∞ (3 + 2/n)/(1 + 1/n²)

3) = (3 + 0)/(1 + 0) = 3

4) The sequence converges to 3.

Problem 2: Series Convergence

Test ∑n=1∞ 1/(n² + 1) for convergence.

Solution:

1) Compare to p-series ∑1/n² (converges since p = 2 > 1)

2) 1/(n² + 1) < 1/n² for all n ≥ 1

3) By Direct Comparison Test, since ∑1/n² converges, ∑1/(n² + 1) also converges

Problem 3: Power Series

Find the radius of convergence for ∑n=0∞ (x-2)ⁿ/n.

Solution:

1) cₙ = 1/n

2) |cₙ₊₁/cₙ| = (1/(n+1)) / (1/n) = n/(n+1)

3) R = limn→∞ |cₙ/cₙ₊₁| = limn→∞ (n+1)/n = 1

4) Radius of convergence is 1

💡 Exam Tips & Strategies

🎯 Problem-Solving Strategy:
  1. Identify the type - sequence, series, or power series
  2. Choose appropriate test - start with simplest tests
  3. Check conditions - ensure test requirements are met
  4. Apply systematically - follow test procedures carefully
  5. State conclusion clearly - convergence or divergence
⚠️ Common Mistakes:
  • Using wrong convergence test
  • Not checking test conditions
  • Confusing sequences and series
  • Incorrect radius of convergence
  • Mixing up Taylor and Maclaurin series
🔑 Key Insight: This unit represents the culmination of calculus. Master the convergence tests and Taylor series, and you'll have the tools to approximate any function and analyze infinite processes.

🧠 Ready to Test Your Knowledge?

📝 Unit 10 Practice Test

Now that you've mastered infinite sequences and series, test your understanding with our comprehensive practice test. The test includes:

  • 15 Multiple Choice Questions - Covering all major topics
  • 6 Free Response Questions - With detailed solutions
  • 45-minute timer - Simulating real exam conditions
  • Instant feedback - Check your answers as you go
  • Score calculation - See how you perform
Take Practice Test

Recommended after completing all unit content

💡 Pro Tip: This is the final unit of AP Calculus BC! Master convergence tests and Taylor series to complete your calculus journey. The practice test will help you identify any remaining gaps before the AP exam.