Section 9.6: Dealing with Negative Terms in Series

Overview

• Previous conditions for tests:
  • Integral Test: positive, continuous, decreasing.
  • Comparison and Limit Comparison Tests: positive terms.
• Problem: How to handle negative terms?
  • Alternating Series Test for alternating positive and negative terms.
  • What if the series does not alternate?

Absolute Convergence

• Definition: Take the absolute value of each term in the series.
• Transform series: |a₁| + |a₂| + ...
• If this series converges, the original series is absolutely convergent.

Example

• Series: (-1)ⁿ⁻¹ / n² (Alternating)
  • Alternating series test shows convergence.
  • Absolute value series: 1/n² (a p-series with p=2 > 1), converges.
  • Hence, original series is absolutely convergent.

Conditional Convergence

• Series can converge but not absolutely.
• Example: Alternating Harmonic Series
  • Converges by alternating series test.
  • Absolute value series: Harmonic Series 1/n, diverges.
  • Series is conditionally convergent (convergent but not absolutely).

Absolute Convergence - Significance

• Stronger than regular convergence.
• If absolute convergence is shown, the series definitely converges.

Classifications

1. Divergent
2. Conditionally Convergent
3. Absolutely Convergent

Series Tests Recap

Divergence Test

• If lim (aₙ) != 0, series diverges.

Geometric Series

• Form: arⁿ
  • Converges if |r| < 1.
  • Diverges if |r| >= 1.
• Sum: S = a / (1 - r)

Telescoping Series

• Use partial fractions to simplify and find limit of partial sums.

P-Series

• Form: 1/nᵖ
  • Converges if p > 1.
  • Diverges if p <= 1.

Integral Test

• Applicable for positive, continuous, decreasing function f(x).
• ∫ from 1 to ∞ convergence implies series convergence.

Comparison and Limit Comparison Tests

• Compare with known convergent/divergent series.

Alternating Series Test

• Conditions:
  1. lim (bₙ) = 0
  2. bₙ is decreasing.
• If both hold, series converges.

Ratio Test

• Use for factorials and powers of n.
• Form: lim |(aₙ₊₁/aₙ)|
  • If < 1, absolutely convergent.
  • If > 1, divergent.
  • If = 1, inconclusive.

Root Test

• Use for nth powers.
• Form: lim ⁿ√|aₙ|
  • If < 1, absolutely convergent.
  • If > 1, divergent.
  • If = 1, inconclusive.

Practical Applications

• Determine appropriate test based on series characteristics.
• Start with divergence test, then check for type.
• Apply comparison for single-term series.
• Use alternating series test for alternating sequences.
• Root or ratio test for sequences with powers or factorials.

Strategy

• Use simpler tests (e.g., divergence, alternating) first.
• For complex structures, try ratio/root tests.
• Absolute value transformations help apply more tests.