講者：許永昌老師 1. Contents Absolute and Conditional Convergence Leibniz criterion (for alternating series) Riemann’s Theorem Convergence Test of conditionally.

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Presentation transcript:

講者：許永昌老師 1

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

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

Benefits of absolutely convergent series 4

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

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

Riemann’s theorem ( 請預讀 P272) 7

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

Homework

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

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

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

Homework

nouns 14