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

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

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

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

$\int _{0}^{\frac {1}{e^{e}}}{\biggl [}(x\uparrow \uparrow \infty )_{Upper}-(x\uparrow \uparrow \infty )_{Lower}{\biggr ]}\cdot dx\approx 0.045405$

Text

$\lim _{Uncertainty\to \infty }\sum _{then}^{now}YourMistakes=Unbearable$

$\lim _{Hate\to \infty }\prod _{lies}^{truth}Anything\;You've\;Said=Ammunition$

$\lim _{Apologies\to Excuses}Familiarity=Contempt$

Daniel Bennett

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

Girls = Evil

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