User:Flippin42/formulæ

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< User:Flippin42

1 e^gamma with products to infinity of kth roots of e
2 nth triangular/n
3 2nd order tetration (n to the n) limit to infinity e relationship
4 nth root of n! limit to 1/e
5 consecutive powers sum
6 Euler's formula
7 Power Towers
8 Text
- 8.1 Daniel Bennett
- 8.2 Girls = Evil

[edit] e^gamma with products to infinity of kth roots of e

$\lim_{n\to\infty} \left ( \prod_{k=1}^{n+1} \sqrt[k]{e} - \prod_{k=1}^{n} \sqrt[k]{e} \right ) = e^\gamma$

[edit] nth triangular/n

$\frac{\displaystyle{\sum_{k=1}^{n+1}} k}{n+1} - \frac{\displaystyle{\sum_{k=1}^n} k}{n} = 0.5$

[edit] 2nd order tetration (n to the n) limit to infinity e relationship

$\lim_{n\to\infty} \left ( \frac{(n+1)^{n+1}}{n^n} - \frac{n^n}{(n-1)^{n-1}} \right ) = e$

[edit] nth root of n! limit to 1/e

$\lim_{n\to\infty} \left ( \sqrt[n+1]{(n+1)!} - \sqrt[n]{n!} \right ) = \frac{1}{e}$

[edit] consecutive powers sum

$\lim_{n\to\infty} \left ( \frac{(n+4)^{n+1} - \underbrace{3^{n+1} + 4^{n+1} + 5^{n+1} + \dotsb}_{n+1}}{(n+3)^n - \underbrace{3^n + 4^n + 5^n + \dotsb}_n} - \frac{(n+3)^n - \underbrace{3^n + 4^n + 5^n + \dotsb}_n}{(n+2)^{n-1} - \underbrace{3^{n-1} + 4^{n-1} + 5^{n-1} + \dotsb}_{n-1}} \right ) = e$

OR...

$\lim_{n\to\infty} \left ( \frac{(n+4)^{n+1} - \displaystyle{\sum_{k=3}^{n+3} k^{n+1}}} {(n+3)^n - \displaystyle{\sum_{k=3}^{n+2} k^n}} - \frac{(n+3)^n - \displaystyle{\sum_{k=3}^{n+2} k^n}}{(n+2)^{n-1} - \displaystyle{\sum_{k=3}^{n+1} k^{n-1}}} \right ) = e$

[edit] Euler's formula

$e^{i\pi} + 1 = 0 \!$

[edit] Power Towers

$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}}}}}}}}}}}}}} \approx 2\!$

$\sqrt{2}^{\overbrace{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}}}}}^\infty} = 2\!$

$\begin{align} \sqrt{2}\uparrow\uparrow\infty &= 2 \\ \sqrt[e]{e} &= 1.444667861... \\ \sqrt[e]{e}\uparrow\uparrow\infty &= e \\ e^{-e} = \frac{1}{e^e} &= 0.065988035... \\ \frac{1}{e^e}\uparrow\uparrow\infty &= \frac{1}{e} \\ 0.001\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.001051251058... \\ 0.992764518... \end{Bmatrix} ; \quad Difference = 0.991713267... \\ 0.01\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.01309252... \\ 0.941488368... \end{Bmatrix} ; \quad Difference = 0.928395848... \\ 0.015\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.021585386... \\ 0.91333526... \end{Bmatrix} ; \quad Difference = 0.891749873... \\ 0.02\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.03146156... \\ 0.884194383... \end{Bmatrix} ; \quad Difference = 0.852732823... \\ 0.03\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.056132967... \\ 0.821327373... \end{Bmatrix} ; \quad Difference = 0.765194406... \\ 0.04\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.08960084... \\ 0.749451269... \end{Bmatrix} ; \quad Difference = 0.659850428... \\ 0.045\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.111117455... \\ 0.708513944... \end{Bmatrix} ; \quad Difference = 0.597396489... \\ 0.05\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.137359395... \\ 0.662660838... \end{Bmatrix} ; \quad Difference = 0.525301443... \\ 0.055\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.170720724... \\ 0.609472066... \end{Bmatrix} ; \quad Difference = 0.438751341... \\ 0.06\uparrow\uparrow\infty &\in \begin{Bmatrix} 0.216898064... \\ 0.54322953... \end{Bmatrix} ; \quad Difference = 0.326331465... \\ \lim_{x\to 0} (x\uparrow\uparrow\infty) &\in \begin{Bmatrix} 0 \\ 1 \end{Bmatrix} \end{align}$

$\int_0^{\frac{1}{e^e}} \biggl [ (x \uparrow\uparrow \infty)_{Upper} - (x \uparrow\uparrow \infty)_{Lower} \biggr ] \cdot dx \approx 0.045405$

[edit] Text

$\lim_{Uncertainty\to\infty} \sum_{then}^{now} Your Mistakes = Unbearable$

$\lim_{Hate\to\infty} \prod_{lies}^{truth} Anything \; You've \; Said = Ammunition$

$\lim_{Apologies\to Excuses} Familiarity = Contempt$

[edit] Daniel Bennett

$\mathfrak{Daniel \; Bennett} \qquad \mathbb{DANIEL \; BENNETT} \qquad \mathcal{DANIEL \quad BENNETT}$

[edit] Girls = Evil

$\begin{align} Girls &= Time \times Money\\ Time &= Money\\ \therefore Girls &= Money^2\\ Money &= \sqrt{Evil}\\ \therefore Girls &= \sqrt{Evil}^2\\ Girls &= Evil \end{align}$