Study Notes on Alternating Series Test and Convergence

Introduction to Alternating Series

• Alternating series typically take the form of a series with terms that alternate signs, commonly represented as (-1)^n a_n.

Alternating Series Test

• Criteria for applying the alternating series test:

  • Criterion 1: The limit as n approaches infinity of bn (where bn = a_n) must equal zero:
  • Example: \lim_{n \to \infty} \frac{1}{n^2} = 0 (Valid, moving on)
  • Criterion 2: The term b{n+1} must be less than bn for all n greater than or equal to 1:
  • Example: Comparing terms:
    • \frac{1}{(n+1)^2} < \frac{1}{n^2} (establishing this through algebra)
    • Upon expanding: \frac{1}{(n+1)^2} < \frac{1}{n^2} leads to valid state as the inequality holds true.

• If both criteria are satisfied, then the series converges by the alternating series test.

Example of Using the Alternating Series Test

• Example Series: \sum_{n=1}^{\infty} (-1)^{n+1} \frac{n}{n+1}
  • Written form (for personal preference): (-1)^{n+1} \frac{n}{n+1}
  • Limit analysis:
  • \lim_{n \to \infty} \frac{n}{n+1} = 1
  • Since this does NOT equal zero, the series diverges.

Remainder Estimation of Alternating Series

• Remainders can be approximated easily with alternating series:
  • Rn \leq b{n+1} is used to estimate how closely the series approximates its sum.
  • To reduce error below a value, for example, \varepsilon = 10^{-5}, find n such that \frac{1}{n+1} < 10^{-5}.
  • Rearranging gives n + 1 > 10^5 or n > 10^5 - 1
  • Implying, the smallest n is 316.

Conditional vs Absolute Convergence

• Conditional Convergence:
  • A series converges conditionally if the series converges but the series of absolute values diverges (
    e.g. \sum{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} to clarify it converges while \sum{n=1}^{\infty} \frac{1}{n} diverges).
• Absolute Convergence:
  • If both the series and its absolute series converge, then the series is absolutely convergent.
• The relationship is crucial, as the behavior of one can dictate conclusions about the other.

Steps to Determine Convergence Type

1. Check the absolute series (ignore the alternating sign):
   -
   1. If the absolute series converges, the original series is absolutely convergent.
   2. If the absolute series diverges, check the alternating series directly to determine conditional convergence or divergence:
   • If it converges, it is conditionally convergent.
   • If it diverges, it is simply divergent.

Theorems Related to Convergence

• If a series is absolutely convergent, it is also conditionally convergent but not vice versa. This emphasizes that checking for absolute convergence first is often beneficial.

Flowchart for Convergence Testing

1. Check the absolute value of the series:
   • Does it converge? If yes, declare absolutely convergent.
   • If no, proceed to step 2.
2. Check the alternating series:
   • Apply the alternating series test (check two criteria).
   • If convergent, declare conditionally convergent.
   • If diverges, declare the overall series as divergent.

Final Remarks on Examples and Practice

• Examples play a vital role in enforcing concepts.
  • Set examples where the target is to establish whether a series absolutely converges, is conditionally convergent, or divergent, typically yielding structured outcomes.
• Regular practice will aid in becoming familiar with identifying the nature of alternating series in various problem contexts.