Divergent Products of Rational Numbers

SUPERVARIETIES

$\displaystyle \prod_{n\geq1} 2^{(-1)^{n-1}} = 2.\dfrac{1}{2}.2.\dfrac{1}{2}.2.\dfrac{1}{2}.2.\dfrac{1}{2}.\ldots = \sqrt{2} \\ = \dfrac{\prod_{n\geq1} 2.1}{\prod_{n\geq1} 1.2} = \dfrac{2.1.2.1.2.1.2.1.\ldots}{1.2.1.2.1.2.1.2.\ldots} = \dfrac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}$ (unstable)

$\displaystyle \prod_{n\geq1} 2^{(-1)^n} = \dfrac{1}{2}.2.\dfrac{1}{2}.2.\dfrac{1}{2}.2.\dfrac{1}{2}.2.\ldots = \dfrac{1}{\sqrt{2}} \\ = \dfrac{\prod_{n\geq1} 1.2}{\prod_{n\geq1} 2.1} = \dfrac{1.2.1.2.1.2.1.2.\ldots}{2.1.2.1.2.1.2.1.\ldots} = \dfrac{\frac{1}{\sqrt{2}}}{1} = \dfrac{1}{\sqrt{2}}$ (unstable)

$\displaystyle \prod_{n\geq1} 2n.\dfrac{1}{2n} = 2.\dfrac{1}{2}.4.\dfrac{1}{4}.6.\dfrac{1}{6}.8.\dfrac{1}{8}\ldots = 1 \\ = \dfrac{\prod_{n\geq1} 2n.1}{\prod_{n\geq1} 1.2n} = \dfrac{2.1.4.1.6.1.8.1.\ldots}{1.2.1.4.1.6.1.8.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{\pi}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{1}{2n}.2n = \dfrac{1}{2}.2.\dfrac{1}{4}.4.\dfrac{1}{6}.6.\dfrac{1}{8}.8.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} 1.2n}{\prod_{n\geq1} 2n.1} = \dfrac{1.2.1.4.1.6.1.8.\ldots}{2.1.4.1.6.1.8.1.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{\pi}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} (2n)^{(-1)^{n-1}} = 2.\dfrac{1}{4}.6.\dfrac{1}{8}.10.\dfrac{1}{12}.14.\dfrac{1}{16}.\ldots = \dfrac{2}{\sqrt{\pi}} \\ = \dfrac{\prod_{n\geq1} (4n\!-\!2).1}{\prod_{n\geq1} 1.4n} = \dfrac{2.1.6.1.10.1.14.1.\ldots}{1.4.1.8.1.12.1.16.\ldots} = \dfrac{2}{\sqrt{\pi}}$ (unstable)

$\displaystyle \prod_{n\geq1} (2n)^{(-1)^n} = \dfrac{1}{2}.4.\dfrac{1}{6}.8.\dfrac{1}{10}.12.\dfrac{1}{14}.16.\ldots = \dfrac{\sqrt{\pi}}{2} \\ = \dfrac{\prod_{n\geq1} 1.4n}{\prod_{n\geq1} (4n\!-\!2).1} = \dfrac{1.4.1.8.1.12.1.16.\ldots}{2.1.6.1.10.1.14.1.\ldots} = \dfrac{\sqrt{\pi}}{2}$ (unstable)

VARIETIES

$\displaystyle \prod_{n\geq1} \dfrac{2n}{2n\!-\!1}.\dfrac{2n}{2n\!-\!1} = 2.2.\dfrac{4}{3}.\dfrac{4}{3}.\dfrac{6}{5}.\dfrac{6}{5}.\ldots = \dfrac{\pi}{2} \\ = \dfrac{\prod_{n\geq1} 2n.2n}{\prod_{n\geq1} (2n\!-\!1).(2n\!-\!1)} = \dfrac{2.2.4.4.6.6.\ldots}{1.1.3.3.5.5.\ldots} = \dfrac{\pi}{2}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n\!-\!1}{2n}.\dfrac{2n\!-\!1}{2n} = \dfrac{1}{2}.\dfrac{1}{2}.\dfrac{3}{4}.\dfrac{3}{4}.\dfrac{5}{6}.\dfrac{5}{6}.\ldots = \dfrac{2}{\pi} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).(2n\!-\!1)}{\prod_{n\geq1} 2n.2n} = \dfrac{1.1.3.3.5.5.\ldots}{2.2.4.4.6.6.\ldots} = \dfrac{2}{\pi}$ (unstable)

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