**The Bell Curve**:

**The Bell Curve**:

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To test if the infinite series converges,

provided the function

in the interval [2, ∞[ is:

- positive &
- decreasing function

you can convert it to

(although they converge to different values.)

**(Traditional Method learnt at O-Level) **** Integration by Parts**:

**NEW Technique taught by this UC-Berkley Chinese Mathematician: “DI” Method**

**First Stop: “D column at 0 **“

**2nd Stop:**

**3rd Stop:**

China 南京航空航天大学 Nanjing University of Aeronautics and Astronautics set the WiFi password as the answer of this integral (first 6 digits).

Can you solve it?

*(If can’t, please revise GCE “A-level” / Baccalaureate / 高考 Calculus* *微积分*)

**Answer** : Break the integral (I) into 2 parts:

I = A(x) + B(x)

**A(x) = – A(-x) => Odd function **

=> **A(x) = 0** since its area canceled out over [-2, 2]

**B(x) = B(-x) => Even function**

Let x = 2 sin t => dx = 2 cos t. dt

x = 2 = 2 sin t => sin t = 1 =>

t =π/ 2x = 0 = 2 sin t => sin t = 0 =>

t = 0

A smarter method using Analytic Geometry: A circle of radius 2 is

**What is Math** ? Interesting article below:

https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/

**Mathematics**= “*that which is learned*“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.

**Difference between Riemann Integral & Lebesgue Integral ?**

Calculus = 微积分

出自: 荀况《荀子.大略》300 BCE [same time as Euclid]

“尽小者大, **积微**者著”

=> 見微知著

This is just a simple but powerful application of Calculus, behind which lies the philosophy of Leibniz:

1. D (=dy/dx) is the inverse function of

2. Calculus Fundamental Technique:

E.g. Sherlock Holmes example:

1. D first:

dT/dt = k(T-Ts)

=> can’t solve directly

2. Take D’s inverse:

=> can solve now !