Lecture on Series Tests: Direct Comparison Test (DCT) and Limit Comparison Test (LCT)

Overview

• Introduction to series tests focusing on Direct Comparison Test (DCT) and Limit Comparison Test (LCT).
• Both tests require positive terms (>0).

Direct Comparison Test (DCT)

• Basic Condition: If (a_n \leq b_n) for all values.
• Convergence:
  • If you know a series (b_n) converges and (a_n \leq b_n), then (a_n) converges.
  • If (b_n) diverges and (a_n \leq b_n), then (a_n) diverges.
• Application: Not dependent on initial terms; can eventually satisfy (a_n \leq b_n).

Limit Comparison Test (LCT)

• Basic Condition: Terms must be positive.
• Limit Condition:
  • Compare ( \lim_{n \to \infty} \frac{a_n}{b_n} = L ), where (L) is finite and positive.
  • If (L = 0), both series converge.
  • If (L = \infty), both series diverge.
• Orientation: Ensure (a_n) is the numerator for certain cases.
• Flexibility: Works if terms become positive eventually.

Application Strategy

• Selection of Series: Choose known series like geometric or p-series.
• Example 1:
  • Series resembles (\frac{1}{n^3}). Known that (\frac{1}{n^3}) converges (p-series, (p = 3 > 1)).
  • Use DCT to show given series converges by proving inequality through cross multiplication.

Example Problems

Example 1: Convergence Comparison

• Known Series: (\frac{1}{n^3}).
• Proving Inequality: Through cross multiplication:
  • Show (n^3(n-1) \leq n^4 + 2).
  • Conclude given series also converges (DCT).

Example 2: Divergence Comparison

• Initial Attempt with DCT:
  • Series resembles (\frac{1}{\sqrt{n}}); p-series (p = \frac{1}{2} < 1) diverges.
  • Attempt to prove (1 \times \sqrt{n} \geq 1 \times (2 + \sqrt{n})). Failed to satisfy.
• Use LCT:
  • Known series: (\frac{1}{\sqrt{n}}) diverges.
  • Compute limit: (\lim_{n \to \infty} \frac{(2 + \sqrt{n})}{\sqrt{n}}).
  • Simplify and compute to find limit = 1.
  • Conclusion: Given series diverges (LCT).

Practice

• Emphasis on practicing comparisons and selecting appropriate series.
• Exercises designed to apply DCT and LCT.

Summary

• Clear strategy and calculation steps essential for determining convergence or divergence.
• DCT and LCT are powerful tools but require practice and careful selection of series for comparison.