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Understanding Series Tests: DCT and LCT

Feb 25, 2025

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.