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e^gamma with products to infinity of kth roots of e[edit]
![{\displaystyle \lim _{n\to \infty }\left(\prod _{k=1}^{n+1}{\sqrt[{k}]{e}}-\prod _{k=1}^{n}{\sqrt[{k}]{e}}\right)=e^{\gamma }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/108853ad849556012be0891ba9b220c2fcadd8ca)
nth triangular/n[edit]
![{\displaystyle {\frac {\displaystyle {\sum _{k=1}^{n+1}}k}{n+1}}-{\frac {\displaystyle {\sum _{k=1}^{n}}k}{n}}=0.5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2452a05c58ad3505387ab44f38197441d7a02b5)
2nd order tetration (n to the n) limit to infinity e relationship[edit]
![{\displaystyle \lim _{n\to \infty }\left({\frac {(n+1)^{n+1}}{n^{n}}}-{\frac {n^{n}}{(n-1)^{n-1}}}\right)=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a10f6afa23770a284680ed1745ac07b7fb662006)
nth root of n! limit to 1/e[edit]
![{\displaystyle \lim _{n\to \infty }\left({\sqrt[{n+1}]{(n+1)!}}-{\sqrt[{n}]{n!}}\right)={\frac {1}{e}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9957e973083a9c2afca5aec0b674e29d98b0e817)
consecutive powers sum[edit]
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59048be6b283c0188ae7da8b5a570157e3690ebb)
OR...
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/294b10ad1fd3b7ef006d706ea719b983e0cda9f1)
Euler's formula[edit]
![{\displaystyle e^{i\pi }+1=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097465363d9ecc8dc765be4ab86ee1f9ba1770a0)
Power Towers[edit]
![{\displaystyle {\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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e840dfa5be3f77c16ff65d35713f81ea3fabbf)
![{\displaystyle {\sqrt {2}}^{\overbrace {{\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\sqrt {2}}}}} ^{\infty }}=2\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1facfa4661b2beb673c7fb1e75d8a9056d61350)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df16eec326594685ffd261c9dcdf9795e0f7ba9f)
![{\displaystyle \int _{0}^{\frac {1}{e^{e}}}{\biggl [}(x\uparrow \uparrow \infty )_{Upper}-(x\uparrow \uparrow \infty )_{Lower}{\biggr ]}\cdot dx\approx 0.045405}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298e2affa4f9108aaca45e0385ea9611d9063218)
![{\displaystyle \lim _{Uncertainty\to \infty }\sum _{then}^{now}YourMistakes=Unbearable}](https://wikimedia.org/api/rest_v1/media/math/render/svg/986b4b53077974e891e8fca2cc48ddfc79c70d90)
![{\displaystyle \lim _{Hate\to \infty }\prod _{lies}^{truth}Anything\;You've\;Said=Ammunition}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60c2e1711ee4df05c84c2cb24ab902d120b22dd9)
![{\displaystyle \lim _{Apologies\to Excuses}Familiarity=Contempt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeaafcf34ade4c86beeccc6a0aae8d1e0aff306f)
Daniel Bennett[edit]
![{\displaystyle {\mathfrak {Daniel\;Bennett}}\qquad \mathbb {DANIEL\;BENNETT} \qquad {\mathcal {DANIEL\quad BENNETT}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b13375a3d2902446d44f508f87fbf9fdf81a3b)
Girls = Evil[edit]
![{\displaystyle {\begin{aligned}Girls&=Time\times Money\\Time&=Money\\\therefore Girls&=Money^{2}\\Money&={\sqrt {Evil}}\\\therefore Girls&={\sqrt {Evil}}^{2}\\Girls&=Evil\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35170fc4ca4cd31ae4414add709d05786f836b8e)