Chapter 9, Section 5: Infinite Series and Alternating Series Test

Key Concepts

• Sequence: A set of numbers described by a function (refer back to Chapter 9, Section 1).
• Series: The sum of a sequence.
• Infinite Series: Adding up an infinite set of numbers; requires special methods to determine convergence.
• Alternating Series: A series where the signs of terms alternate between positive and negative values.

Alternating Series Test

• Alternating series: Terms switch between negative and positive.
• Types of alternation:
  • Odd terms are negative, even terms are positive.
  • Even terms are negative, odd terms are positive.

Conditions for Convergence

1. First Condition: Limit of the sequence (a_n) as (n) approaches infinity must be 0.
2. Second Condition: Each term (a_{n+1}) must be smaller than or equal to the preceding term (a_n)._

Applying the Test

• Use ((-1)^n) or ((-1)^{n+1}) to identify alternating series.
• Simplify the series by focusing on the magnitude of terms without alternating signs.
• Use the limits and conditions to determine if the series converges.

Example Calculation

• Take the limit of (1/n) as (n) approaches infinity: result is 0.
• Prove that (1/n \geq 1/(n+1)) by cross-multiplying to show (n \leq n+1).
• Conclusion: If conditions are met, the series converges.

When the Test Fails

• If the limit of (a_n) does not equal 0, the test immediately fails.
• Example: ((-1)^{n+1} \cdot (n+1)/n) does not satisfy condition 1.
• Use L'Hôpital's Rule for limits that give indeterminate forms.

Alternating Series Remainder Theorem

• S: True sum of the series.
• S_n: Current partial sum up to (n).
• R_n: Remainder to reach the true sum.
• a_{n+1}: Value of the next term.
• The remainder is always ≤ the value of the next term, providing an upper bound for error._

Steps to Use the Theorem

1. Calculate (S_n) by summing the first (n) terms.
2. Identify the next term (a_{n+1}).
3. Assert that ( R_n \leq a_{n+1} ).
4. Estimate the true sum (S) within bounds of (S_n +/- a_{n+1})._

Practical Application: Error Tolerance

• Determine the number of terms needed for a desired error tolerance.
• Example: Require error < 0.001, solve (1/(n+1)^4 < 0.001) to find (n).
• Use algebra to solve for (n) and ensure sufficient terms are added.

Summary

• Alternating series test helps in determining convergence.
• Understand when the test fails and use other methods.
• Use the alternating series remainder for approximate sums and error estimation.
• Ensure calculations and conditions are clearly justified in solutions.