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 “
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 / 2
x = 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:
- 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]
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 !
Why Newton’s Calculus Not Rigorous?
cancel x (≠0)from upper and below =>
In : we assume x ≠ 0, so cancel upper & lower x
But In : assume x=0 to get L=5
 (x ≠ 0) contradicts with  (x = 0)
This is the weakness of Newtonian Calculus, made rigorous later by Cauchy’s ε-δ ‘Analysis’.