# Quod Erat Demonstrandum

## 2015/09/13

### 一式過

Filed under: Pure Mathematics — johnmayhk @ 5:19 下午
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$-1,1,-1,1,\dots$

$(-1)^n$

## 2014/05/21

### 與漸近線相交

Filed under: Fun,Pure Mathematics — johnmayhk @ 1:35 下午
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「人工」地使圖像產生「劇烈」變化， (more…)

## 2014/02/11

### 二項極限

Filed under: NSS,Pure Mathematics — johnmayhk @ 3:16 下午
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（圖片來源：Mathematics）

“…is a recently discovered beauty, probably in 2012…"

## 2010/05/11

### 無聊中四數學課紀錄：(1 + 1/n)^n

Filed under: NSS,Teaching — johnmayhk @ 10:57 下午
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$e = \displaystyle\lim_{n \rightarrow \infty}(1 + \frac{1}{n})^n$

(a) $\displaystyle\lim_{n \rightarrow \infty}(1 + \frac{1}{2n})^{2n} = ?$ (more…)

## 2009/07/03

### 計到即存在?

Filed under: HKALE,Pure Mathematics — johnmayhk @ 8:41 上午
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$\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}}$ = ?

## 2009/06/04

### 溫書題

Filed under: HKALE,HKCEE,mathematics,Pure Mathematics — johnmayhk @ 4:28 下午
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1. 有關數列的題目

=======================================
Let {$a_n$} be a sequence of positive integers. Define sequences {$b_n$} and {$c_n$} as
$b_1 = a_1, b_2 = a_1a_2 + 1, b_{n+2} = a_{n+2}b_{n+1} + b_{n}$. ($n \in \mathbb{N}$)
$c_1 = 1, c_2 = a_2, c_{n+2} = a_{n+2}c_{n+1} + c_{n}$. ($n \in \mathbb{N}$)
Let $x_n = \frac{b_n}{c_n}$. ($n \in \mathbb{N}$)

Show that $x_1 \le \lim_{n \rightarrow \infty}x_n \le 1 + x_1$.
======================================= (more…)

1961 (more…)

## 2009/04/22

### 一些有關不能使用 l’Hôpital’s rule 的例子

Filed under: HKALE,Pure Mathematics — johnmayhk @ 9:31 上午
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## 2009/04/17

### 可導性

Filed under: HKALE,Pure Mathematics — johnmayhk @ 5:23 下午
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1. Put $x = 0$ into the expression of $f'(x)$ and it is undefined, then $f(x)$ is not differentiable at $x = 0$. True?
2. If $\lim_{x \rightarrow 0^-}f'(x) = \lim_{x \rightarrow 0^+}f'(x)$, then $f(x)$ is differentiable at $x = 0$. True?
3. If $\lim_{x \rightarrow 0^-}f'(x)$ is finite, then the value of $\lim_{h \rightarrow 0^-}\frac{f(h) - f(0)}{h}$ is also finite. True? (more…)

## 2009/03/29

### 大數值的乘階

$1! = 1$
$2! = 1 \times 2 = 2$
$3! = 1 \times 2 \times 3 = 6$
$4! = 1 \times 2 \times 3 \times 4 = 24$

$170! \approx 7.2574 \times 10^{306}$

## 2009/03/10

### Limit of tan(x)/x

Filed under: HKALE,Pure Mathematics — johnmayhk @ 9:42 下午
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When introducing the following ‘important’ limit to F.6B boys,

$\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1$,

$\lim_{x \rightarrow \infty} \frac{\sin x}{x}$ = ? (more…)