Calculus II - Alternating Series Test

Introduction

• Alternating series contain terms that alternate in sign.
• Important for series with negative terms.

Forms of Alternating Series

• Terms can be expressed as:
  • ( a_n = (-1)^n b_n )
  • ( a_n = (-1)^{n+1} b_n )

Alternating Series Test

• Conditions for convergence:
  1. (\lim_{n \to \infty} b_n = 0)
  2. ({b_n}) is a decreasing sequence eventually.
• Note:
  • This test only checks for convergence, not divergence.
  • Decreasing condition only needs to hold eventually._

Why the Test is Valid

• Series can split into finite and infinite parts:
  • Finite: ( \sum_{n=1}^{N} (-1)^n b_n )
  • Infinite: ( \sum_{n=N+1}^{\infty} (-1)^n b_n )
• Convergence depends on the infinite series.
• Finite series sum is finite; thus adding to convergent infinite series results in convergence.

Examples

1. Alternating Harmonic Series:
   • Example of a series where terms naturally decrease.
2. More Complex Series:
   • Example with terms needing careful analysis for decreasing nature.
3. Further Examples:
   • Series with more complex expressions need detailed work to show decreasing nature.

Proof of Alternating Series Test

• Example proof for ( a_n = (-1)^{n+1} b_n ).
• ( {s_{2n}} ) is an increasing sequence:
  • Series of even partial sums increases and is bounded.
• Limit of even and odd partial sums equal:
  • ( \lim_{n \to \infty} s_{2n+1} = \lim_{n \to \infty} s_{2n} = s ).
• Conclusion: Series ( a_n ) is convergent if both sequences converge to the same limit._

Conclusion

• Alternating series test is crucial for determining convergence in series with alternating signs.
• Ensure conditions for convergence are verified, especially the eventual decrease of terms.