# Quod Erat Demonstrandum

## 2010/11/02

### 開放日開放數學

Filed under: Fun,Pure Mathematics,School Activities — johnmayhk @ 3:22 下午

$\frac{1}{998999} = 0.000001001002003005008013021034055...$

http://www.wolframalpha.com/input/?i=1%2F998999

$1,1,2,3,5,8,13,21,34,55,\dots$

$0.000001001002003005008013021034055...$

$\frac{1}{10^6} + \frac{1}{10^9} + \frac{2}{10^{12}} + \frac{3}{10^{15}} + \frac{5}{10^{18}} + \dots$

$\displaystyle\sum_{k=0}^\infty \frac{a_k}{10^{3+3k}}$

$a_0 = 0$
$a_1 = 1$
$a_{n+2} = a_{n+1} + a_n$ （其中 $n \ge 0$

$\frac{1}{\sqrt{5}}(\alpha^n - \beta^n)$

（注 1：修純數或 M2 的同學，試用（比方說 M.I.）證之。）
（注 2：$\alpha, \beta$$x^2 - x - 1 = 0$ 的根。）

$\displaystyle\sum_{k=0}^\infty \frac{a_k}{10^3+3k}$

$= \frac{1}{10^3\sqrt{5}}\displaystyle\sum_{k=0}^\infty \frac{1}{10^{3k}}(\alpha^k - \beta^k)$

$= \frac{1}{10^3\sqrt{5}}\displaystyle\sum_{k=0}^\infty ((\frac{\alpha}{10^3})^k - (\frac{\beta}{10^3})^k)$

$= \frac{1}{10^3\sqrt{5}}(\frac{1}{1 - (\alpha/10^{3})} - \frac{1}{1 - (\beta/10^{3})})$

$= \frac{1}{\sqrt{5}}(\frac{1}{10^3 - \alpha} - \frac{1}{10^3 - \beta})$

$= \frac{1}{\sqrt{5}}(\frac{\alpha - \beta}{(10^3 - \alpha)(10^3 - \beta)})$

$= \frac{1}{\sqrt{5}}\frac{\sqrt{5}}{10^6 - 10^3(\alpha + \beta) + \alpha\beta}$

$= \frac{1}{1000000 - 1000 - 1}$

$= \frac{1}{998999}$

$\pi = 2\times\frac{2 \times 2}{1 \times 3} \times \frac{4 \times 4}{3 \times 5} \times \frac{6 \times 6}{5 \times 7} \times \dots$

（這個容易，考慮 $\int_{0}^{\pi/2} \sin^n\theta d\theta$ 便可證明，也順道看看：

$1 + \frac{1}{1 \times 3} + \frac{1}{1 \times 3 \times 5} + \frac{1}{1 \times 3 \times 5 \times 7} + \dots = \sqrt{\frac{\pi e}{2}}$

http://en.wikipedia.org/wiki/Srinivasa_Ramanujan