1. 講者： 許永昌 老師 1

2. Contents Absolute and Conditional Convergence Leibniz criterion (for alternating series) Riemann’s Theorem Convergence Test of conditionally convergent series Multiplication of Series Product Convergence Theorem 2

3. Absolute and Conditional Convergence ( 請預讀 P271) 3  u n,or  u(n)dn  |u n |,or  |u(n)|dn Absolutely convergent  Conditionally convergent  X

4. Benefits of absolutely convergent series 4

5. Leibniz criterion (for alternating series) ( 請預讀 P270) 5

6. Example: ( 請預讀 P271) Alternating Harmonic Series: Because  1/n diverges, it is not an absolutely convergent series. Based on Leibniz criterion, it is convergent. If we rearrange the terms based on a rule: Taking positive terms until the partial sum are equal to or greater than 3/2, then adding in negative terms until the partial sum just fall below 3/2, and so on. It will converge to 3/2  ln2. ???? Why ???? 6

7. Riemann’s theorem ( 請預讀 P272) 7

8. Convergence Test of conditionally convergent series ( 請預讀 P272~P273) 8

9. Homework 5.3.2 9

10. Multiplication of Series ( 請預讀 P275) The standard or Cauchy product of two series 10 n m n l Rearrangement

11. Example: The multiplication of conditional series may be divergent. 11

12. Product Convergent Theorem ( 請 預讀 P275) 12 u0u0 u1u1 u3u3 u4u4 We get

13. Homework 5.4.1 13

14. nouns 14