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

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