# Eikonal Blog

## 2010.02.09

### Products

• $\prod_{n=1}^\infty \frac{(2n)^2}{(2n-1)(2n+1)} = \frac\pi2$, (John Wallis; 1616-1703)
• $\prod_{n=1}^\infty \frac{(4n)^2}{(4n-1)(4n+1)} = \frac{\pi\sqrt{2}}4$,
• $\prod_{n=1}^\infty \frac{(4n+2)^2}{(4n+1)(4n+31)} = \sqrt{2}$,
• Question: what is the value of $P_m:\equiv \prod_{n=1}^\infty \frac{(mn)^2}{(mn-1)(mn+1)}$? Clearly $P_2= \frac\pi2$ and $P_4= \frac{\pi\sqrt{2}}4$.
• $\prod_{n=1}^\infty \left(1+\frac{(-)^{n+1}}{2n-1}\right) = \sqrt{2}$,
• $\prod_{n=1}^\infty \left(1-\frac1{n^2}\right) = \frac12$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^2}\right) = \frac{\sinh(\pi)}\pi = 3.0\cdots$,
• $x \prod_{n=1}^\infty \left(1- \frac{x^2}{n^2\pi^2}\right) = \sin(x)$,
• $x \prod_{n=1}^\infty \left(1+ \frac{x^2}{n^2\pi^2}\right) = \sinh(x)$,
• $\prod_{n=1}^\infty \left(1- \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cos(x)$,
• $\prod_{n=1}^\infty \left(1+ \frac{4x^2}{(2n-1)^2\pi^2}\right) = \cosh(x)$,
• $\prod_{n=3}^\infty \left(1-\frac4{n^2}\right) = \frac16$,
• $\prod_{n=1}^\infty \left(1+\frac1{n^3}\right) = \frac{\cosh(\\pi\sqrt{3}/2)}\pi = 2.428,\cdots$,
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^s}\right) = \frac1{\zeta(s)}$ (Leonard Euler), where ${\Bbb P}$ is the set of prime numbers, and $\zeta(s):\equiv\sum_{n=1}^\infty \frac1{n^s}$ is the Riemann’s zeta function.
• $\prod_{p\in{\Bbb P}} \left(1-\frac1{p^2}\right) = \frac6{\pi^2}$.

Strange forms:

• $\frac{\sqrt{2}}2 \frac{\sqrt{2+\sqrt{2}}}2 \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}2 \cdots = \frac{2}{\pi}$, (Francois Viete; 1540-1603).
• $\sqrt{x \cdot \sqrt{x\cdot \sqrt{x \cdots}}} = x$,
• $\sqrt[n+1]{x \cdot \sqrt[n+1]{x \cdots}} = \sqrt[n]{x}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} \frac1n & 1 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 0 & e \\ 0 & 1\end{pmatrix}$,
• $\prod_{n=1}^\infty$ $\begin{pmatrix} 1 & \frac1{n!} \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 1 & e \\ 0 & 1\end{pmatrix}$, (which is quite trivial).

Sources:

• Calvin C. Clawson “Mathematical Sorcery” (Revealing the secrets of numbers); ISBN 0-7382-0496-X; Perseus Publishing