User:Flippin42/formulæ

=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

$$

=nth triangular/n=



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

$$

=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

$$

=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}

$$

=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

$$

=Euler's formula=



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

$$

=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

$$

=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

$$

Daniel Bennett


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

$$

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}

$$