My Intended Trigo Answer

The Math.SE question $2\cos(2x) - 2\sin(x) = 0$ has attracted several answers from high-rep users.

I am expanding @rhombic's comment into an answer.

\[ \begin{aligned} 2\cos(2x)-2\sin(x)&=0 \\ 2 - 4\sin^2(x)-2\sin(x)&=0 \\ 2\sin^2(x)+\sin(x) - 1&=0 \\ (2 \sin(x) - 1)(\sin(x) +1) &= 0 \\ \sin(x) = \frac12 \text{ or } \sin(x) &= -1 \\ x = \frac{\pi}{6}, \frac{5\pi}{6} \text{ (rejected) or } & \frac{3\pi}{2} \text{ (rejected)} \end{aligned} \]

JavaScript Copy Button

Goal To create a copy button for my Math.SE comment template in order to save the trouble of copying and pasting. My first attempt I put the boilerplate inside a Markdown codeblock to prevent them from getting interpreted by Hugo’s Markdown parser. Under each codeblock, I placed the copy button. Comment boilerplate goes here ... πŸ“ Another comment boilerplate goes here ... πŸ“ … My page’s original layout [Read More]

My Dual Answer

Update: The question has been reopened. I intended to answer κΉ€μ’…ν˜„'s problem on Math.SE. However, the programs in the question body aren't typeset in MathJax. As a result, I downvoted and closed this question because found it unclear. From the proposed dual, it seems that I shouldn't interpret the primal as a linear program. Anyways, without further clarifications from OP, I found no reason to look at this further. [Read More]

Borel Cantelli Exercise 2019

I intend to post this for a Borel-Cantelli lemma exercise on Math.SE.

The target event is ${\exists i_0 \in \Bbb{N} : \forall i \ge i_0, X_i = 1}$, whose complement is

\[ \{\forall i_0 \in \Bbb{N} : \exists i \ge i_0, X_i > = 0\} = \limsup_i \{X_i = 0\}. \]

To apply Borel-Cantelli, one has to determine whether $\sum_i P(X_i = 0)<+\infty$.

Weak LLN Practice

My intended answer to a weak LLN problem on Math.SE. Problem: Suppose $(X_n)$ is a sequence of r.v's satisfying $P(X_n=\pm\ln (n))=\frac{1}{2}$ for each $n=1,2\dots$. I am trying to show that $(X_n)$ satisfies the weak LLN. The idea is to show that $P(\overline{X_n}>\varepsilon)$ tends to 0, but I am unsure how to do so. My solution: As in the accepted answer in OP's previous question, I'll assume the independence of $(X_n)$. [Read More]

Solution to a $p$-test Exercise

I intended to answer Maddle's $p$-test question, but T. Bongers has beaten me by two minutes, so I posted my answer here to save my work. The problem statement This is the sum: \(\sum\limits_{n=3}^\infty\frac{1}{n\cdot\ln(n)\cdot\ln(\ln(n))^p}\) How do I tell which values of $p$ allow this to converge? The ratio test isn't working out for me at all. Unpublished solution The integral test will do. \[ \begin{aligned} & \int_3^{+\infty} \frac{1}{x\cdot\ln(x)\cdot\ln(\ln(x))^p} \,dx \\ &= \int_3^{+\infty} \frac{1}{\ln(x)\cdot\ln(\ln(x))^p} \,d(\ln x) \\ &= \int_3^{+\infty} \frac{1}{\ln(\ln(x))^p} \,d(\ln(\ln(x))) \\ &= \begin{cases} [\ln(\ln(\ln(x)))]_3^{+\infty} & \text{if } p = 1 \\ \left[\dfrac{[\ln(\ln(x))}{p+1}]^{p+1} \right]_3^{+\infty} & \text{if } p \ne 1 \end{cases} \end{aligned} \] [Read More]