# Quod Erat Demonstrandum

## 2009/03/09

### Just answer questions from F7 student

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 2:45 下午

Q.1

sin x 同cos x 既taylor’s theorem expansion say the last term of sin x is (-1)^(n-1)*x^(2n-1)/(2n-1)! 個power of x of the remainder term 係咪2n or 2n+1 都得? similarly, the last term of cos x is (-1)^n*x^(2n)/(2n)! 個power of x of the remainder term 係咪2n+1 or 2n+2 都得?

$\sin x$ 的 Taylor’s expansion 是

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$

$\sin x \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + \frac{x^{2n - 1}}{(2n - 1)!}$

$\frac{f^{(2n + 1)}(\xi)}{(2n + 1)!}x^{2n + 1}$ (其中 $\xi$ 介乎 0 和 $x$ 之間)

$\frac{f^{(2n)}(\xi)}{(2n)!}x^{2n}$ 作為餘項？

$\sin x \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + \frac{x^{2n - 1}}{(2n - 1)!}$ – – – (1)

$\sin x \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots + \frac{x^{2n - 1}}{(2n - 1)!} + 0$ – – – (2)

Open-ended question：對任何函數，取泰勒展式（Taylor’s expansion）後，是否取愈多項，誤差（餘項）的絕對值一定愈小？

Q.2

revision notes p.22 theorem 2 如果將there exists a positive constant K s.t. max of absolute f'(x)=K<1 改做x-g(x) monotonic 係咪只可以PROVE 到unique sol. 而converge 就唔得?

1. $g(x) \in [0,1]$ for all $x \in [0,1]$
2. $g(x) - x$ is monotonic decreasing