Lecture on Series Tests: Direct and Limit Comparison Tests

Overview

• Introduction to two series tests:
  • Direct Comparison Test (DCT)
  • Limit Comparison Test (LCT)
• Both tests are applied to series with positive terms.

Direct Comparison Test (DCT)

• Concept: Used to determine convergence or divergence by comparing a series ( a_n ) with another series ( b_n ).
• Condition: ( a_n \leq b_n ) for all terms.
• Application:
  • If ( b_n ) converges and ( a_n \leq b_n ), then ( a_n ) converges.
  • If ( a_n ) diverges and ( a_n \leq b_n ), then ( b_n ) diverges.
• Modification: The inequality ( a_n \leq b_n ) only needs to hold eventually, not necessarily for initial terms.

Limit Comparison Test (LCT)

• Concept: Compares the limits of ratios of terms to a known series.
• Procedure:
  • Let ( L ) be the limit of the ratio ( a_n/b_n ).
  • If ( L ) is finite and positive, both series converge or diverge.
  • Special cases:
    • ( L = 0 ): Both series converge.
    • ( L = \infty ): Both series diverge.
• Flexibility: Works regardless of which term is on top, ( a_n ) or ( b_n ), as long as ( L ) is finite and positive.

Choosing Series for Comparison

• Important to pick series like geometric or p-series with known convergence/divergence properties.
• Practice is essential to master the selection of comparison series.

Examples and Practice

Example 1

• Series Considered: ( 1/n^3 ), a known convergent p-series (( p = 3 > 1 )).
• Comparison:
  • Establish ( 1/n^3 ) as smaller.
  • Use cross-multiplication to verify inequality.
  • Conclude convergence using DCT.

Example 2

• Initial Attempt: Compare with ( 1/\sqrt{n} ), a divergent p-series (( p = 1/2 < 1 )).
• Inequality Check: Failed to prove using DCT.
• Switch to LCT:
  • Calculate limit of the ratio.
  • Find limit to be 1 (finite and positive), indicating series diverges like the comparison series.
• Conclusion: Divergence using LCT.

Conclusion

• Practice various tests as the choice of test depends on the series.
• Both DCT and LCT are essential tools in understanding series convergence/divergence.
• Always confirm known series properties (geometric, p-series, etc.) before comparison.