My Intended Trigo Answer
Posted on May 1, 2019
(Last modified on May 2, 2019)
 1 minutes
 Vincent Tam

0 comment
The Math.SE question $2\cos(2x)  2\sin(x) = 0$ has attracted several
answers from highrep 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
Posted on April 26, 2019
(Last modified on April 27, 2019)
 3 minutes
 Vincent Tam

0 comment
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 ... π
Another comment boilerplate goes here ... π
…
My page’s original layout
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My Dual Answer
Posted on April 26, 2019
(Last modified on April 29, 2019)
 2 minutes
 Vincent Tam

1 comment
Update: The question has been reopened.
I intended to answer κΉμ’
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.
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Borel Cantelli Exercise 2019
Posted on February 4, 2019
(Last modified on April 28, 2019)
 1 minutes
 Vincent Tam

1 comment
I intend to post this for a BorelCantelli 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 BorelCantelli, one has to determine whether $\sum_i P(X_i =
0)<+\infty$.
Weak LLN Practice
Posted on December 2, 2018
(Last modified on April 28, 2019)
 1 minutes
 Vincent Tam

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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 https://math.stackexchange.com/q/3021650/290189, I'll assume the independence of $(X_n)$.
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Solution to a $p$test Exercise
Posted on November 20, 2018
(Last modified on April 28, 2019)
 1 minutes
 Vincent Tam

0 comment
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.
\[ \begin{aligned} & \int_3^{+\infty} \frac{1}{x\cdot\ln(x)\cdot\ln(\ln(x))^p} \,dx \\ &= \int_3^{+\infty} \frac{1}{\ln(x)\cdot\ln(\ln(x))^p} \,d(\ln x) \\ &= \int_3^{+\infty} \frac{1}{\ln(\ln(x))^p} \,d(\ln(\ln(x))) \\ &= \begin{cases} [\ln(\ln(\ln(x)))]_3^{+\infty} & \text{if } p = 1 \\ \left[\dfrac{[\ln(\ln(x))}{p+1}]^{p+1} \right]_3^{+\infty} & \text{if } p \ne 1 \end{cases} \end{aligned} \]
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