TheSimpliFire

$$( ͡° ͜ʖ ͡°)\cdots\LaTeX\,\textsf{is fun}$$

https://mathscmp.wordpress.com

Chatroom: TheSimpliFire's Chatroom!

Discussions/answers to conjectures (may or may not be original), interesting maths problems in/out of MSE, and any random stuff.

Anyone with a maths interest level of at least $|z|$ with $\Re z\ne0$ is more than welcome :)

You are welcome if $\Re z=\Im z=0$.


My WordPress blog is here!

About: Where I store any mathematical problems I've made up :)


My favourite questions:

  • What is the maximum value of $\sqrt{\frac{x}{x+\sqrt{\frac{x^2}{x-\sqrt{\frac{x^3}{x+ \sqrt{ \frac{x^4}{x-\cdots}}}}}}}}$?

  • On the integral $\int_{-\pi/2}^{\pi/2}\sin(x/\sin(x/\sin(x/\sin\cdots)))\,dx$ and On the integral $\int_0^\pi\sin(x+\sin(x+\sin(x+\cdots)))\,dx$

  • Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

  • Some interesting observations on a sum of reciprocals

  • How to show that $\int_0^\infty\frac1{x^x}\,dx<2$

  • Mathematical coincidences concerning the numbers $\pi$, $e$ and $163$

  • Convergence concerning the $\alpha$th derivative of $f(x)=x^{\alpha}-\alpha^x$

My favourite answers:

  • Prove that $a^{ab}+b^{bc}+c^{cd}+d^{da}\ge\pi$

  • Evaluating $\int_0^1\frac{3x^4+ 4x^3 + 3x^2}{(4x^3 + 3x^2 + 2x+ 1)^2}\, dx$

  • How to prove :$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <3$

  • Evaluate $\int (1-x^{2008})^{\frac{1}{2007}} (1-x^{2007})^{\frac{1}{2008}} dx$

  • General formula for arctan

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