[微積分日記] CH11

[shmm (shmm的暱稱)]

寶哥要我po 系花要我po

所以先po一點 希望能拋磚引玉啦

先來我所知道的 ch11

Chapter 11 Infinite Series

p.644 Theorem 11.1.6

          ∞
      if  Σ ak converges, then ak -> 0 as k -> ∞
          k=0

      ! converges, diverges與起始點無關

      ! 證明"iff"時要順證也要反證

p.648 # 判別 converges, diverges

      * <1> ak -> 0 as k -> ∞

    *** <2> Integral Test

         ** (a) Basic Comparison

        *** (b) Limit Comparison

        <3> *** (a) Root Test (次方)  ┐
                                      ├ 與 1 有關
            *** (b) Ratio Test (階乘) ┘

      ! Σf(k) 可用 ∫f(k) dk 來比

p.652 ! e^k >> k >> lnk

p.653 **** Theorem 11.2.6 The Limit Comparison Test

      孔:"一定要很熟練"

      [30sec] Theorem 11.2.6 的 proof

p.657 Theorem 11.3.1 The Root Test

      # The Root Test

       (ak)^(1/k) -> ρ

       ρ<1, converges

       ρ>1, diverges

       ρ=1, use "Limit Comparison"

p.658 Theorem 11.3.2 The Ratio Test

      # The Ratio Test

       a(k+1)
       ——— -> r
         ak

       r>1, diverges

       r<1, converges

       r=1 use "Limit Comparison"

      ! 想成等比級數來理解

      [10sec] EX3

p.662 Definition 11.4.3 Conditional Convergence

      ! 不可經過加減乘除

p.663 Theorem 11.4.4 Alternating Series Test

      # Alternating Series Test

       (a) 後項 < 前項

       (b) ak -> 0

      [30sec] Theorem 11.4.4 的 proof

p.665 [10sec] EX5

p.668 # Taylor Polynomials

       用一高次多項式,來盡量表示一函式
                    (k)
               ∞  f   (0)
       Pn(x) = Σ ———— x^k
               k=0   k!

       孔:"非常重要呀 寫下來"

p.671 Theorem 11.5.1 Taylor's Theorem

      ! Rn+1(x) 可視為誤差

p.674 ! 問 Taylor Series 可以不必知道 remainder

p.680 # 有3種方法

       1. Po(x) = g(a)

       2. Integeration by part

       3. 移軸法 [30sec]  孔:"最簡單 但是觀念要清楚"

p.683 (11.6.5) 孔:"ln0 = ∞ 不可以呀"

p.684 ! Power Series 是在求得 radiu convegagentive

p.686 Figure 11.7.1

      ! 端點收斂不確定

p.689 [20sec] EX6

p.694 [10sec] EX2

p.695 [10sec] EX3

p.704 [10sec] Supplement To Section 11.8

       PS. 阿海後面因為那一堂之前2天沒睡 所以該堂啄龜

           故都只有記到秒數 希望有記的人回po吧

       -=給別人方便就是給自己方便=-


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