Calculus Cheatsheet: Mastering Sequences and Series Convergence Tests

Introduction

Sequences and series convergence tests are essential tools in calculus that help determine whether infinite sums approach a finite value. Understanding these tests is crucial for analyzing functions, solving differential equations, and approximating complex expressions. This cheatsheet provides a comprehensive reference for the most important convergence tests and their applications.

Core Concepts of Sequences and Series

Sequences

• A sequence is an ordered list of numbers: {a₁, a₂, a₃, …} or {aₙ}
• A sequence converges if lim(n→∞) aₙ exists and equals a finite value
• A sequence diverges if the limit doesn’t exist or equals ±∞

Series

• A series is the sum of terms in a sequence: a₁ + a₂ + a₃ + … or Σaₙ
• Convergent series: The sum approaches a finite value as n→∞
• Divergent series: The sum grows without bound or oscillates
• Partial sum: Sₙ = a₁ + a₂ + … + aₙ
• A series converges if lim(n→∞) Sₙ exists and equals a finite value

Convergence Tests for Series

1. Divergence Test (nth Term Test)

• Statement: If lim(n→∞) aₙ ≠ 0, then Σaₙ diverges
• Contrapositive: If Σaₙ converges, then lim(n→∞) aₙ = 0
• Note: If lim(n→∞) aₙ = 0, the series may still diverge
• Example: Harmonic series Σ(1/n) diverges despite lim(n→∞) 1/n = 0

2. Geometric Series Test

• Form: Σarⁿ⁻¹ from n=1 to ∞
• Convergence: Converges if |r| < 1
• Sum: If |r| < 1, then Σarⁿ⁻¹ = a/(1-r)
• Example: Σ(1/2)ⁿ converges to 1/(1-1/2) = 2

3. p-Series Test

• Form: Σ(1/nᵖ) from n=1 to ∞
• Convergence:
  • Converges if p > 1
  • Diverges if p ≤ 1
• Examples:
  • Σ(1/n²) converges (p = 2 > 1)
  • Σ(1/n) diverges (p = 1 ≤ 1)

4. Comparison Test

• Statement: If 0 ≤ aₙ ≤ bₙ for all n ≥ N:
  • If Σbₙ converges, then Σaₙ converges
  • If Σaₙ diverges, then Σbₙ diverges
• Best used: When series terms can be bounded by a known series

5. Limit Comparison Test

• Statement: If aₙ, bₙ > 0 and lim(n→∞) aₙ/bₙ = c where c > 0:
  • Σaₙ and Σbₙ either both converge or both diverge
• Best used: When two series behave similarly as n→∞
• Example: Compare Σ(n/(n²+1)) with Σ(1/n)

6. Ratio Test

• Statement: Let L = lim(n→∞) |aₙ₊₁/aₙ|
  • If L < 1: Series converges absolutely
  • If L > 1: Series diverges
  • If L = 1: Test is inconclusive
• Best used: For series with factorials or exponentials
• Example: Σ(2ⁿ/n!) converges as L = lim(n→∞) 2/(n+1) = 0 < 1

7. Root Test

• Statement: Let L = lim(n→∞) ⁿ√|aₙ|
  • If L < 1: Series converges absolutely
  • If L > 1: Series diverges
  • If L = 1: Test is inconclusive
• Best used: When aₙ involves powers like nⁿ
• Example: Σ(1/nⁿ) converges as L = lim(n→∞) (1/n) = 0 < 1

8. Integral Test

• Statement: If f(x) is positive, continuous, and decreasing for x ≥ N, and aₙ = f(n), then:
  • Σaₙ converges if and only if ∫ₙ^∞ f(x)dx converges
• Best used: When aₙ can be represented as a continuous function
• Example: For Σ(1/n²), use f(x) = 1/x² and ∫₁^∞ (1/x²)dx = 1

9. Alternating Series Test (Leibniz Test)

• Form: Σ(-1)ⁿ⁻¹bₙ or Σ(-1)ⁿbₙ where bₙ > 0
• Convergence: Converges if:
  • bₙ₊₁ ≤ bₙ for all n ≥ N (decreasing sequence)
  • lim(n→∞) bₙ = 0
• Error bound: |Sₙ – S| ≤ bₙ₊₁
• Example: Σ(-1)ⁿ⁻¹(1/n) converges (alternating harmonic series)

10. Absolute Convergence Test

• Statement: If Σ|aₙ| converges, then Σaₙ converges
• Note: A series can converge conditionally (Σaₙ converges but Σ|aₙ| diverges)
• Best used: After using ratio or root test on |aₙ|
• Example: Σ(-1)ⁿ⁻¹(1/n) converges conditionally because Σ(1/n) diverges

11. Direct Comparison Test

• Statement: If 0 ≤ aₙ ≤ bₙ for all n ≥ N:
  • If Σbₙ converges, then Σaₙ converges
  • If Σaₙ diverges, then Σbₙ diverges
• Example: Compare Σ(1/(n²+n)) with Σ(1/n²)

12. Cauchy Condensation Test

• Statement: For a non-increasing sequence aₙ > 0, Σaₙ converges if and only if Σ2ᵏa₂ᵏ converges
• Best used: For series where comparison is difficult
• Example: For Σ(1/n), check Σ2ᵏ(1/2ᵏ) = Σ1, which diverges

Comparison Table of Convergence Tests

Common Challenges and Solutions

Challenge 1: Choosing the Right Test

• Solution: Follow a systematic approach:
  1. Check if lim(n→∞) aₙ = 0 (if not, series diverges)
  2. Identify series type (geometric, alternating, etc.)
  3. For positive series, try comparison, ratio, or integral tests
  4. For alternating series, check the alternating series test first

Challenge 2: Dealing with Complex Terms

• Solution:
  • Break into simpler components
  • Use algebraic manipulations
  • Apply limits to simplify expressions
  • Example: For Σ(n²+1)/(n³+n), simplify to Σ(1/n + 1/n³) as n→∞

Challenge 3: Inconclusive Tests

• Solution:
  • If ratio/root test gives L = 1, try another test
  • Combine multiple tests when necessary
  • Example: For Σ(1/n·ln(n)), ratio test is inconclusive, but integral test works

Challenge 4: Determining Absolute vs. Conditional Convergence

• Solution:
  • First check if Σ|aₙ| converges (absolute convergence)
  • If Σ|aₙ| diverges but series contains alternating signs, check for conditional convergence
  • Example: Σ(-1)ⁿ⁻¹/n is conditionally convergent

Best Practices and Tips

Tip 1: Simplification Strategy

• Always simplify terms before applying tests
• Use asymptotic behavior (what happens as n→∞)
• Focus on highest-order terms for large n
• Example: Σ(n²+n)/(n³+5) ≈ Σ(n²/n³) = Σ(1/n) as n→∞

Tip 2: Test Selection Guide

• For series with factorials: Ratio Test
• For series with nth powers: Root Test
• For alternating series: Alternating Series Test
• For positive series comparable to known series: Comparison Tests
• For series expressible as a function: Integral Test

Tip 3: Common Convergent Series

• Σ(1/n²) converges to π²/6
• Σ(1/n(n+1)) converges to 1
• Σ(r^n) converges to 1/(1-r) for |r| < 1
• Σ((-1)ⁿ⁻¹/n) converges to ln(2)

Tip 4: Common Divergent Series

• Σ(1/n) (harmonic series)
• Σ(1/n^p) for p ≤ 1
• Σ(n/(n+1))
• Σ(1/ln(n))

Resources for Further Learning

Textbooks

• Calculus by James Stewart
• Thomas’ Calculus by George B. Thomas
• Advanced Calculus by Lynn H. Loomis and Shlomo Sternberg

Online Resources

• Khan Academy: Sequences and Series
• MIT OpenCourseWare: Single Variable Calculus
• Paul’s Online Math Notes: Calculus II
• 3Blue1Brown YouTube channel for visual intuition

Practice Resources

• Brilliant.org (interactive calculus problems)
• Wolfram Alpha (verify solutions)
• PatrickJMT YouTube videos for worked examples

Quick Reference: Series Convergence Flowchart

1. Apply Divergence Test

   • If lim(n→∞) aₙ ≠ 0 → DIVERGES
   • If lim(n→∞) aₙ = 0 → CONTINUE TESTING

2. Check series type:

   • If geometric series (ar^(n-1)) → CONVERGES if |r| < 1
   • If p-series (1/n^p) → CONVERGES if p > 1
   • If alternating → Try Alternating Series Test

3. For positive series:

   • Try Ratio Test
   • Try Root Test
   • Try Comparison Tests
   • Try Integral Test

4. If alternating:

   • Check absolute convergence first
   • If absolutely convergent → CONVERGES
   • If not, check Alternating Series Test conditions