User:Wonghang

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It is User:wonghang. I am a high school student in Hong Kong.

1 My Introduction
2 Pi formula proofs
3 External Links

[edit] My Introduction

I am a high school student in Hong Kong. I am 0x13 years old. I was born at 0x1e88ccc8. I scored 38 marks in AQ Test. I know that I am a computer geek. You can read my geek code as follow:

-----BEGIN GEEK CODE BLOCK-----
Version: 3.12
GCS>CS/E/M/P/S/MU dpu s: a--->? C+++ ULB++++ P L+>+++++ E>+++ W++ 
N+ o? K? w++>+++++ O-- M- V-- PS+ !PE Y+ PGP++>+++ t?
5? X? R+++>- tv+ b++>++++ DI? D++ G++>+++ e->++++ h!>++
r--->+++ y--
------END GEEK CODE BLOCK------

I am interested in Computer, Mathematics and Physics.

I like Canotopop including but not limited to Twins and Eason Chan. I also like listening to some Classical music such as Liebestraume No. 3, Pachelbel's Canon.

I want to be a computer scientist in the future.

[edit] Pi formula proofs

[edit] Proof of Leibniz's formula

Proof of Leibniz formula

[edit] Proof of Wallis's product

Proof of Wallis product

[edit] Proof of ζ(2)

Basel problem

[edit] Proof of Machin's formula

Machin's formula

$\frac{\pi}{4} = 4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}$

[edit] Proof

Recall the formulas:

$\tan (x+y) = \frac{\tan x + \tan y}{1- \tan x \tan y}$

$\tan (x-y) = \frac{\tan x - \tan y}{1+ \tan x \tan y}$

$\tan (2x) = \frac{2 \tan x}{1 - \tan^2 x}$

Let

$\tan \alpha = \frac{1}{5}$

We can obtain tan(2α) = 5/12 and tan(4α) = 120/119 by using the above formula. Therefore,

$\tan ( 4 \tan^{-1} \frac{1}{5} ) = \frac{120}{119}$

Consider,

$\tan (4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239}) = \frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{119} \frac{1}{239}} = \frac{120 \cdot 239 - 119}{119 \cdot 239 + 120} = \frac{120(120 + 119) - 119}{119(120 + 119) + 120} = \frac{120^2 + 119^2}{120^2 + 119^2} = 1$