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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.
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