Joshua AC's Blogspot

This post will be about several proofs of random theorems or formulas that I enjoy proving and I would like to share them with everyone. I hope that everyone could see the beauty of mathematical proofs from this post. Formula 1 For a positive integer $n$, it always holds true that: [\lim_{x \to 0} \frac{1-\prod_{i=1}^n cos(a_i x)}{x^2} = \frac{1}{2} \cdot \sum_{i=1}^n a_i^2] Note for some who might not be used to the notation, $\prod$ here is just like $\sum$ but instead of summation of terms, it is the product of all the terms. For example $\prod_{i=1}^3 i = 1 \cdot 2 \cdot 3 = 6$ Simple Example: [\lim_{x \to 0} \frac{1-cos3x \cdot cos4x \cdot cos5x}{x^2} = \frac{3^2+4^2+5^2}{2} = 25] Proof: In dealing with problem where there exist $n$ term products, we should try to build it from the most simple case of $n=1$, and then try to prove for $n=k+1$ using the equality from $n=k$ (similar to a domino effect ). In mathematics, we call this method as  induction . First step i...

In this post, I would like to write a solution to an integral problem (possibly more problems in the future) as well as give insights that are usually not taught, in spite of its importance. Without any longer, let us delve into the problem. Problem 1 Find the value of the integral: [ \int \frac{\sqrt{4x-x^2}}{x} dx ] Insight 1 First of all, in dealing with integrals, we should usually try to avoid forms with square roots . Moreover, it is usually harder to manipulate the equation if we have multiple $x$ terms in a square root. Thus, in this problem, the most troublesome part is $\sqrt{4x-x^2}$. After knowing this, several ideas to simplify this form might come up.  The first most common ways to deal with $x$ and $x^2$ terms is by completing the square . In this case, $4x-x^2$ can be written as $-(x-2)^2+4$. However, the problem with this approach is that the denominator is $x$, so if we want to do a substitution of $u=x-2$, the denominator would be in the form of $u+2$, which...