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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