SUNY Geneseo Department of Mathematics

Absolute Convergence and Related Convergence Tests

Thursday, April 2

Math 222 01
Spring 2015
Prof. Doug Baldwin

Misc
- Infinite series problem set
Questions?
Comparison Tests
- Sum of (n+3) / (n²-5)
Convergence in Series with Mixed Signs
- Section 10.5
  - Absolute convergence test
    - If sum of |a_n| converges, then sum a_n converges
  - Ratio test
    - lim_n→∞ | a_n+1 / a_n | = p
    - p < 1: converges
    - p = 1: ???
    - p > 1 (or ∞): diverges
  - Root test
    - lim_n→∞ (|a_n|)^1/n = p
    - p < 1: converges
    - p = 1: ???
    - p > 1 (or ∞): diverges
- Examples
  - Sum of (sin n) / n²
  - Sum of n2ⁿ / n!
  - Sum of n2ⁿ / 3ⁿ
    - Dealing formally with limits such as (n+1)/n?
  - Sum of n^e / 3ⁿ
Next
- Alternating series
- Read section 10.6 (“Alternating Series and Conditional Convergence”)