SEQUENCES & SERIES

SEQUENCES & SERIES

1. S Summation of series

   • ①S是 summation of series 符号. ②当然了, ∑ 可以来表示. start from r=1 OR 0 OR #, ended by n sometimes.
   • Applying standard formulas: Σr=1/2(n)(n+1), Σr^2=1/6(n)(n+1)(2n+1), Σr^3=(Σr)^2
     • 等式的证明: ∑r^3=(∑r)^2--> (1^3+2^3+...+n^3)=(1+2+3+...+n)^2
   • Method of differences: 追击问题
     • *do not reduce fractions to their lowest terms, since this obscures the cancellation which should occur.  --保持原样, 不要直接就化简.
     • * 注意S 的起点和终点. e.g. 有∑, start from r=0. 
     • 化简: listing 一般首3~4项+尾3~4项, 消除得到几个项. *注意化简答案 common factor(公因数).
       • Q: 到底要列几项才能找到用来消除的term? '追击问题'. 他是有规律可循的: e.g. ∑(r+1)/2 - (r-1)/2. 你知道他 (r+1) 和 (r-1) 在互相追击.  (1+1) - (1-1). (2+1) - (2-1). (3+1) - (3-1)→追到了. (r-1)+1+1 既第三项的 (r-1) 追到了第一项的 (r+1). 可以消除.
   • Partial functions: term种类/ common factors个数
     • e.g. ①(x+1)/ [(x-1)(x-2)^2=A/ (x-1) + B/ (x+2) + C/ (x+2)^2. ②(2x^2 - 3)=(√2x+√3)(√2x-√3)...
       • 1.=>如果写成别种,就无意义...= =

2. M&T series: a linear approximation for f(x) (when x is close to 0/ a)

   • f(x)=... it is likely an approximation. but also a 'equation'. 
   • 解题: 代换 OR 已有general,找每一项.
   • Maclaurin's seires: Power series in x: f(x)=f(0)+xf'(0)+x^2/2f''(0)+x^3/6f'''(0)+...=∑x^n/(n!) f^n[(a)]
   • Taylor series: Power series in (x-a): f(x)=f(a)+(x-a)f'(a)+(x-a)^2/2 f''(a)+(x-a)^3/6 f'''(a)+......
     • App1. Find a Taylor series: f(x)=..., f'(x)=..., f''(x)=...代值进去--> f(x)=......
     • App2. Use a Tylor series to solve differential equations: since that also comes in series??
     • App3. Series expansions of compound fuctions: 转化为乘法, 分式可以表示(...)(...)^-1