Formulas - Revision history

Admin: Add Binet's Fibonacci Formula (via update-page on MediaWiki MCP Server)

2026-05-25T21:23:55Z

Add Binet's Fibonacci Formula (via update-page on MediaWiki MCP Server)

← Older revision		Revision as of 21:23, 25 May 2026
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	</math>		</math>

			Binet's Fibonacci Formula [https://en.wikipedia.org/wiki/Fibonacci_sequence#Binet's_formula]:

			<math>
			F_n = \dfrac{ \varphi^n - \psi^n }{ \sqrt{5} }
			</math>

			where <math>\varphi = \dfrac{1 + \sqrt{5}}{2}</math> (the golden ratio) and <math>\psi = \dfrac{1 - \sqrt{5}}{2} = -\varphi^{-1}</math>.

	{{MathFlag}}		{{MathFlag}}

Admin: Add Wallis Product for Pi and Stirling's Approximation (via update-page on MediaWiki MCP Server)

2026-05-25T21:22:22Z

Add Wallis Product for Pi and Stirling's Approximation (via update-page on MediaWiki MCP Server)

← Older revision		Revision as of 21:22, 25 May 2026
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	<math>		<math>
	\pi^{-1} = \displaystyle{ \dfrac{\sqrt{8}}{99^2} } \displaystyle{ \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } } \displaystyle{ \dfrac{1103 + 26390 k}{99^{4k}} }		\pi^{-1} = \displaystyle{ \dfrac{\sqrt{8}}{99^2} } \displaystyle{ \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } } \displaystyle{ \dfrac{1103 + 26390 k}{99^{4k}} }
			</math>

			Wallis Product for Pi [https://en.wikipedia.org/wiki/Wallis_product]:

			<math>
			\dfrac{\pi}{2} = \prod_{n=1}^{\infty} \left( \dfrac{2n}{2n-1} \cdot \dfrac{2n}{2n+1} \right) = \dfrac{2}{1} \cdot \dfrac{2}{3} \cdot \dfrac{4}{3} \cdot \dfrac{4}{5} \cdot \dfrac{6}{5} \cdot \dfrac{6}{7} \cdots
	</math>		</math>

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	<math>		<math>
	\sum_{n=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}		\sum_{n=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}
			</math>

			Stirling's Approximation [https://en.wikipedia.org/wiki/Stirling%27s_approximation]:

			<math>
			n! \sim \sqrt{2 \pi n} \left( \dfrac{n}{e} \right)^n
	</math>		</math>

Admin at 21:59, 7 March 2019

2019-03-07T21:59:38Z

← Older revision		Revision as of 21:59, 7 March 2019
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	</math>		</math>

	Euler formula:		Euler formula [https://www.math.purdue.edu/~eremenko/dvi/euler.pdf]:

	<math>		<math>

Admin at 21:57, 7 March 2019

2019-03-07T21:57:37Z

← Older revision		Revision as of 21:57, 7 March 2019
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	<math>		<math>
	e^{i \pi} + 1 = 0		e^{i \pi} + 1 = 0
			</math>

			Euler's sum of inverse squares:

			<math>
			\sum_{n=1}^{\infty} \dfrac{1}{n^2} = \dfrac{\pi^2}{6}
	</math>		</math>

Admin at 21:55, 7 March 2019

2019-03-07T21:55:15Z

← Older revision		Revision as of 21:55, 7 March 2019
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	<math>		<math>
	\sum_{k=1}^{\infty} k = - \frac{1}{12}		\sum_{k=1}^{\infty} k = - \frac{1}{12}
	~~</math>~~

	~~Zeta-regularized product:~~

	~~<math>~~
	~~\prod_{k=1}^{\infty} k = \sqrt{2 \pi}~~
	</math>		</math>

Admin at 08:15, 4 March 2019

2019-03-04T08:15:22Z

← Older revision		Revision as of 08:15, 4 March 2019
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	b_{2n} = \sqrt{a_{2n} b_n}		b_{2n} = \sqrt{a_{2n} b_n}
	</math>		</math>


			{{MathFlag}}

Admin at 08:14, 4 March 2019

2019-03-04T08:14:29Z

← Older revision		Revision as of 08:14, 4 March 2019
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	<math>		<math>
	\pi^{-1} = \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{99^{4k}}		\pi^{-1} = \displaystyle{ \dfrac{\sqrt{8}}{99^2} } \displaystyle{ \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } } \displaystyle{ \dfrac{1103 + 26390 k}{99^{4k}} }
	</math>		</math>

Admin at 08:13, 4 March 2019

2019-03-04T08:13:27Z

← Older revision		Revision as of 08:13, 4 March 2019
Line 1:		Line 1:
	The most beautiful formulas:		The most beautiful formulas:

	~~Gauss~~'s integral of		Ramanujan's inverse pi formula:

			<math>
			\pi^{-1} = \dfrac{\sqrt{8}}{99^2} \sum_{k \geq 0} \dfrac{ (4k)! }{ (4^k k!)^4 } \dfrac{1103 + 26390 k}{99^{4k}}
			</math>

			Gaussian integral:

	<math>		<math>

Admin: Created page with "The most beautiful formulas: Gauss's integral of $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{2 \pi}$ Ramanujan sum: $\sum_{k=1}^{\infty} k = - \frac{1..."$

2019-03-04T08:09:47Z

Created page with "The most beautiful formulas: Gauss's integral of <math> \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{2 \pi} </math> Ramanujan sum: <math> \sum_{k=1}^{\infty} k = - \frac{1..."

New page

The most beautiful formulas:

Gauss's integral of

<math>
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{2 \pi}
</math>

Ramanujan sum:

<math>
\sum_{k=1}^{\infty} k = - \frac{1}{12}
</math>

Zeta-regularized product:

<math>
\prod_{k=1}^{\infty} k = \sqrt{2 \pi}
</math>

Euler formula:

<math>
e^{i \pi} + 1 = 0
</math>

Archimedes' Recurrence Formula:

<math>
a_{2n} = \frac{2 a_n b_n}{a_n + b_n}
</math>

<math>
b_{2n} = \sqrt{a_{2n} b_n}
</math>