# Quod Erat Demonstrandum

## 2018/03/14

### 黃金比某級數

Filed under: Fun,mathematics,NSS — johnmayhk @ 11:11 上午
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$\displaystyle \Phi=\frac{1}{\Phi}+\frac{1}{\Phi^2}+\frac{1}{\Phi^3}+\dots$

$\displaystyle \Phi=\frac{1+\sqrt{5}}{2}$

(more…)

## 2015/08/28

### sum of 1/k^2 from 1 to infinity

Filed under: Additional / Applied Mathematics,Fun — johnmayhk @ 4:30 下午
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…………………………………………….

$ax^n+bx^{n-1}+\dots +cx+d=0$ ………. (*)

$-\frac{b}{a}$(more…)

## 2008/10/10

### Say something about series in Applied Mathematics (II)

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 4:53 下午
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Here is just a typical question in AL Applied Mathematics (II).

For natural numbers $m, n$ ($m \ge n$).

Let $f(x) = x^ne^x$, evaluate $f^{(m)}(0)$. (more…)

## 2008/09/29

### Just a question of applied math. from a F.7 boy

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 6:13 下午
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Just discuss an easy AL Applied Mathematics (II) Question with students.

Let $f(x), g(x), h(x)$ be twice differentiable functions such that $f(x) = g^2(x) + x^3h(x)$.

(a) Let $p(x) = \frac{f(x)}{g(x)}$, where $g(0) \ne 0$. Show that $p(0) = g(0), p'(0) = g'(0), p''(0) = g''(0)$.

(b) Using (a), or otherwise, find Taylor’s expansion of the function $\frac{2x^4 - 3x^3 + x + 4}{\sqrt{x + 4}}$ about $x = 0$, up to the term in $x^2$. (more…)