Divergent Products of Rational Numbers

HYPERSPECIES

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n}{2n\!-\!1} = 2.\dfrac{4}{3}.\dfrac{6}{5}.\dfrac{8}{7}.\dfrac{10}{9}.\dfrac{12}{11}.\ldots = \sqrt{\pi} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n}{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!-\!1)} = \dfrac{2.4.6.8.10.12.\ldots}{1.3.5.7.\,\;9.11.\ldots} = \dfrac{\frac{\sqrt{\pi}}{2^k}}{\frac{1}{2^k}} = \sqrt{\pi}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n}{2n\!-\!1}.1 = 2.1.\dfrac{4}{3}.1.\dfrac{6}{5}.1.\dfrac{8}{7}.1.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n.1}{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!-\!1).1} = \dfrac{2.1.4.1.6.1.8.1.\ldots}{1.1.3.1.5.1.7.1.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.\dfrac{2n}{2n\!-\!1} = 1.2.1.\dfrac{4}{3}.1.\dfrac{6}{5}.1.\dfrac{8}{7}.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2n}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.(2n\!-\!1)} = \dfrac{1.2.1.4.1.6.1.8.\ldots}{1.1.1.3.1.5.1.7.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n\!-\!1}{2n} = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! \left(1-\dfrac{1}{2n}\right) = \dfrac{1}{2}.\dfrac{3}{4}.\dfrac{5}{6}.\dfrac{7}{8}.\dfrac{9}{10}.\dfrac{11}{12}.\ldots = \dfrac{1}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!-\!1)}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n} = \dfrac{1.3.5.7.\,\;9.11.\ldots}{2.4.6.8.10.12.\ldots} = \dfrac{\frac{1}{2^k}}{\frac{\sqrt{\pi}}{2^k}} = \dfrac{1}{\sqrt{\pi}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n\!-\!1}{2n}.1 = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! \left(1-\dfrac{1}{2n}\right)\!.1 = \dfrac{1}{2}.1.\dfrac{3}{4}.1.\dfrac{5}{6}.1.\dfrac{7}{8}.1.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!-\!1).1}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n.1} = \dfrac{1.1.3.1.5.1.7.1.\ldots}{2.1.4.1.6.1.8.1.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.\dfrac{2n\!-\!1}{2n} = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! 1.\!\left(1-\dfrac{1}{2n}\right) = 1.\dfrac{1}{2}.1.\dfrac{3}{4}.1.\dfrac{5}{6}.1.\dfrac{7}{8}.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.(2n\!-\!1)}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2n} = \dfrac{1.1.1.3.1.5.1.7.\ldots}{1.2.1.4.1.6.1.8.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n}{2n\!+\!1} = \dfrac{2}{3}.\dfrac{4}{5}.\dfrac{6}{7}.\dfrac{8}{9}.\dfrac{10}{11}.\dfrac{12}{13}.\ldots = \dfrac{\sqrt{\pi}}{2} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n}{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!+\!1)} = \dfrac{2.4.6.8.10.12.\ldots}{3.5.7.9.11.13.\ldots} = \dfrac{\frac{\sqrt{\pi}}{2^k}}{\frac{1}{2^{k-1}}} = \dfrac{\sqrt{\pi}}{2}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n}{2n\!+\!1}.1 = \dfrac{2}{3}.1.\dfrac{4}{5}.1.\dfrac{6}{7}.1.\dfrac{8}{9}.1.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n.1}{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!+\!1).1} = \dfrac{2.1.4.1.6.1.8.1.\ldots}{3.1.5.1.7.1.9.1.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.\dfrac{2n}{2n\!+\!1} = 1.\dfrac{2}{3}.1.\dfrac{4}{5}.1.\dfrac{6}{7}.1.\dfrac{8}{9}.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}} \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2n}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.(2n\!+\!1)} = \dfrac{1.2.1.4.1.6.1.8.\ldots}{1.3.1.5.1.7.1.9.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n\!+\!1}{2n} = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! \left(1+\dfrac{1}{2n}\right) = \dfrac{3}{2}.\dfrac{5}{4}.\dfrac{7}{6}.\dfrac{9}{8}.\dfrac{11}{10}.\dfrac{13}{12}.\ldots = \dfrac{2}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!+\!1)}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n} = \dfrac{3.5.7.9.11.13.\ldots}{2.4.6.8.10.12.\ldots} = \dfrac{\frac{1}{2^{k-1}}}{\frac{\sqrt{\pi}}{2^k}} = \dfrac{2}{\sqrt{\pi}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \dfrac{2n\!+\!1}{2n}.1 = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! \left(1+\dfrac{1}{2n}\right)\!.1 = \dfrac{3}{2}.1.\dfrac{5}{4}.1.\dfrac{7}{6}.1.\dfrac{9}{8}.1.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} (2n\!+\!1).1}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 2n.1} = \dfrac{3.1.5.1.7.1.9.1.\ldots}{2.1.4.1.6.1.8.1.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (semi-stable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.\dfrac{2n\!+\!1}{2n} = \prod_{\substack{n\geq1\\k \text{ shifts}}} \! 1.\!\left(1+\dfrac{1}{2n}\right) = 1.\dfrac{3}{2}.1.\dfrac{5}{4}.1.\dfrac{7}{6}.1.\dfrac{9}{8}.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi}} \\ \\ = \dfrac{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.(2n\!+\!1)}{\prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2n} = \dfrac{1.3.1.5.1.7.1.9.\ldots}{1.2.1.4.1.6.1.8.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (semi-stable)

SUPERSPECIES

$\displaystyle \prod_{n\geq1} 2^{-\frac{1-(-1)^n}{2}} = \prod_{n\geq1} \dfrac{1}{2}.1 = \dfrac{1}{2}.1.\dfrac{1}{2}.1.\dfrac{1}{2}.1.\dfrac{1}{2}.1.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} 1.1}{\prod_{n\geq1} 2.1} = \dfrac{1.1.1.1.1.1.1.1.\ldots}{2.1.2.1.2.1.2.1.\ldots} = \dfrac{1}{1} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} 2^{-\frac{1+(-1)^n}{2}} = \prod_{n\geq1} 1.\dfrac{1}{2} = 1.\dfrac{1}{2}.1.\dfrac{1}{2}.1.\dfrac{1}{2}.1.\dfrac{1}{2}.\ldots = \sqrt{2} \\ = \dfrac{\prod_{n\geq1} 1.1}{\prod_{n\geq1} 1.2} = \dfrac{1.1.1.1.1.1.1.1.\ldots}{1.2.1.2.1.2.1.2.\ldots} = \dfrac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}$ (unstable)

$\displaystyle \prod_{n\geq1} (2n\!-\!1).\dfrac{1}{2n\!+\!1} = 1.\dfrac{1}{3}.3.\dfrac{1}{5}.5.\dfrac{1}{7}.7.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).1}{\prod_{n\geq1} 1.(2n\!+\!1)} = \dfrac{1.1.3.1.5.1.7.1.\ldots}{1.3.1.5.1.7.1.9.\ldots} = \dfrac{\sqrt{2}}{\sqrt{2}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} n^{(-1)^{n-1}} = 1.\dfrac{1}{2}.3.\dfrac{1}{4}.5.\dfrac{1}{6}.7.\dfrac{1}{8}.\ldots = \sqrt{\dfrac{2}{\pi}} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).1}{\prod_{n\geq1} 1.2n} = \dfrac{1.1.3.1.5.1.7.1.\ldots}{1.2.1.4.1.6.1.8.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (unstable)

$\displaystyle \prod_{n\geq1} n^{(-1)^n} = 1.2.\dfrac{1}{3}.4.\dfrac{1}{5}.6.\dfrac{1}{7}.8.\ldots = \sqrt{\dfrac{\pi}{2}} \\ = \dfrac{\prod_{n\geq1} 1.2n}{\prod_{n\geq1} (2n\!-\!1).1} = \dfrac{1.2.1.4.1.6.1.8.\ldots}{1.1.3.1.5.1.7.1.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2}}$ (unstable)

$\displaystyle \prod_{n\geq1} n^{2(-1)^{n-1}} = 1^2.\dfrac{1}{2^2}.3^2.\dfrac{1}{4^2}.5^2.\dfrac{1}{6^2}.7^2.\dfrac{1}{8^2}.\ldots = \left(\!\sqrt{\dfrac{2}{\pi}}\right)^2 \!= \dfrac{2}{\pi} \\ = \dfrac{\prod_{n\geq1} n^{(-1)^{n-1}}}{\prod_{n\geq1} n^{(-1)^n}} = \dfrac{1^2.1.3^2.1.5^2.1.7^2.1.\ldots}{1.2^2.1.4^2.1.6^2.1.8^2.\ldots} = \dfrac{(\sqrt{2})^2}{(\sqrt{\pi})^2} = \dfrac{2}{\pi}$ (unstable)

SPECIES

$\displaystyle \prod_{n\geq1} n^{(-1)^{n-1}} = \prod_{n\geq1} (2n\!-\!1).\dfrac{1}{2n} = 1.\dfrac{1}{2}.3.\dfrac{1}{4}.5.\dfrac{1}{6}.\ldots = \sqrt{\dfrac{2}{\pi}} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).1}{\prod_{n\geq1} 1.2n} = \dfrac{1.1.3.1.5.1.7.1.\ldots}{1.2.1.4.1.6.1.8.\ldots} = \dfrac{\sqrt{2}}{\sqrt{\pi}}$ (unstable)

$\displaystyle \prod_{n\geq1} (2n\!-\!1).\dfrac{1}{2n\!-\!1} = 1.1.3.\dfrac{1}{3}.5.\dfrac{1}{5}.7.\dfrac{1}{7}.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).1}{\prod_{n\geq1} 1.(2n\!-\!1)} = \dfrac{1.1.3.1.5.1.7.1.\ldots}{1.1.1.3.1.5.1.7.\ldots} = \dfrac{\sqrt{2}}{\sqrt{2}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} n.\dfrac{1}{n} = 1.1.2.\dfrac{1}{2}.3.\dfrac{1}{3}.4.\dfrac{1}{4}.\ldots = \dfrac{1}{\sqrt{2}} \\ = \dfrac{\prod_{n\geq1} n.1}{\prod_{n\geq1} 1.n} = \dfrac{1.1.2.1.3.1.4.1.\ldots}{1.1.1.2.1.3.1.4.\ldots} = \dfrac{\sqrt{\pi}}{\sqrt{2\pi}} = \dfrac{1}{\sqrt{2}}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{n}{2}.\dfrac{2}{n} = \dfrac{1}{2}.2.1.1.\dfrac{3}{2}.\dfrac{2}{3}.2.\dfrac{1}{2}.\dfrac{5}{2}.\dfrac{2}{5}.\ldots = \dfrac{1}{2} \\ = \dfrac{\prod_{n\geq1} n.2}{\prod_{n\geq1} 2.n} = \dfrac{1.2.2.2.3.2.4.2.\ldots}{2.1.2.2.2.3.2.4.\ldots} = \dfrac{\sqrt{\dfrac{\pi}{2}}}{\sqrt{2\pi}} = \dfrac{1}{2}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{n}{n\!+\!1} = \prod_{n\geq2} \left(1-\dfrac{1}{n}\right) = \dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}.\dfrac{4}{5}.\dfrac{5}{6}.\dfrac{6}{7}.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} n}{\prod_{n\geq1} (n\!+\!1)} = \dfrac{1.2.3.4.5.6.\ldots}{2.3.4.5.6.7.\ldots} = \dfrac{\sqrt{2\pi}}{\sqrt{2\pi}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{n\!+\!1}{n} = \prod_{n\geq1} \left(1+\dfrac{1}{n}\right) = 2.\dfrac{3}{2}.\dfrac{4}{3}.\dfrac{5}{4}.\dfrac{6}{5}.\dfrac{7}{6}.\ldots = 1 \\ = \dfrac{\prod_{n\geq1} (n\!+\!1)}{\prod_{n\geq1} n} = \dfrac{2.3.4.5.6.7.\ldots}{1.2.3.4.5.6.\ldots} = \dfrac{\sqrt{2\pi}}{\sqrt{2\pi}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n\!-\!1}{2} = \prod_{n\geq1} \left(n-\dfrac{1}{2}\right) = \dfrac{1}{2}.\dfrac{3}{2}.\dfrac{5}{2}.\dfrac{7}{2}.\dfrac{9}{2}.\dfrac{11}{2}.\ldots = \sqrt{2} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1)}{\prod_{n\geq1} 2} = \dfrac{1.3.5.7.9.11.\ldots}{2.2.2.2.2.\;\,2.\ldots} = \dfrac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n}{2n\!-\!1} = \prod_{n\geq1} \left(1+\dfrac{1}{2n\!-\!1}\right) = 2.\dfrac{4}{3}.\dfrac{6}{5}.\dfrac{8}{7}.\dfrac{10}{9}.\dfrac{12}{11}.\ldots = \sqrt{\pi} \\ = \dfrac{\prod_{n\geq1} 2n}{\prod_{n\geq1} (2n\!-\!1)} = \dfrac{2.4.6.8.10.12.\ldots}{1.3.5.7.\;\,9.11.\ldots} = \dfrac{\sqrt{\pi}}{1} = \sqrt{\pi}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n}{2n\!+\!1} = \prod_{n\geq2} \left(1-\dfrac{1}{2n\!-\!1}\right) = \dfrac{2}{3}.\dfrac{4}{5}.\dfrac{6}{7}.\dfrac{8}{9}.\dfrac{10}{11}.\dfrac{12}{13}.\ldots = \dfrac{\sqrt{\pi}}{2} \\ = \dfrac{\prod_{n\geq1} 2n}{\prod_{n\geq1} (2n\!+\!1)} = \dfrac{2.4.6.8.10.12.\ldots}{3.5.7.9.11.13.\ldots} = \dfrac{\sqrt{\pi}}{2}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n\!-\!1}{2n\!+\!1} = \prod_{n\geq1} \left(1-\dfrac{2}{2n\!+\!1}\right) = \dfrac{1}{3}.\dfrac{3}{5}.\dfrac{5}{7}.\dfrac{7}{9}.\dfrac{9}{11}.\dfrac{11}{13}.\ldots = \dfrac{1}{2} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1)}{\prod_{n\geq1} (2n\!+\!1)} = \dfrac{1.3.5.7.\;\,9.11.\ldots}{3.5.7.9.11.13.\ldots} = \dfrac{1}{2}$ (unstable)

$\displaystyle \prod_{n\geq2} \dfrac{n^2\!-\!1}{n^2} = \prod_{n\geq2} \left(1-\dfrac{1}{n^2}\right) = \dfrac{3}{4}.\dfrac{8}{9}.\dfrac{15}{16}.\dfrac{24}{25}.\dfrac{35}{36}.\dfrac{48}{49}.\ldots = \dfrac{1}{2} \\ \prod_{n\geq1} \dfrac{n-1}{n}.\dfrac{n+1}{n} = \dfrac{1}{2}.\dfrac{3}{2}.\dfrac{2}{3}.\dfrac{4}{3}.\dfrac{3}{4}.\dfrac{5}{4}.\dfrac{4}{5}.\dfrac{6}{5}.\ldots = \dfrac{1}{2} \\ = \dfrac{\prod_{n\geq2} (n-1) \,.\, \prod_{n\geq2} (n+1)}{\prod_{n\geq2} n \,.\, \prod_{n\geq2} n} = \dfrac{1.3.2.4.3.5.4.6.\ldots}{2.2.3.3.4.4.5.5.\ldots} = \dfrac{1}{2}$ (convergent)

$\displaystyle \prod_{n\geq2} \dfrac{n^2}{n^2\!-\!1} = \prod_{n\geq2} \left(1+\dfrac{1}{n^2-1}\right) = \dfrac{4}{3}.\dfrac{9}{8}.\dfrac{16}{15}.\dfrac{25}{24}.\dfrac{36}{35}.\dfrac{49}{48}.\ldots = 2 \\ \prod_{n\geq1} \dfrac{n}{n-1}.\dfrac{n}{n+1} = 2.\dfrac{2}{3}.\dfrac{3}{2}.\dfrac{3}{4}.\dfrac{4}{3}.\dfrac{4}{5}.\dfrac{5}{4}.\dfrac{5}{6}.\ldots = 2 \\ = \dfrac{\prod_{n\geq2} n \,.\, \prod_{n\geq2} n}{\prod_{n\geq2} (n-1) \,.\, \prod_{n\geq2} (n+1)} = \dfrac{2.2.3.3.4.4.5.5.\ldots}{1.3.2.4.3.5.4.6.\ldots} = 2$ (convergent)

$\displaystyle \prod_{n\geq1} \dfrac{n}{2n\!-\!1}.\dfrac{n}{2n\!-\!1} = 1.1.\dfrac{2}{3}.\dfrac{2}{3}.\dfrac{3}{5}.\dfrac{3}{5}.\ldots = \dfrac{\pi}{\sqrt{2}} \\ = \dfrac{\prod_{n\geq1} n.n}{\prod_{n\geq1} (2n\!-\!1).(2n\!-\!1)} = \dfrac{1.1.2.2.3.3.\ldots}{1.1.3.3.5.5.\ldots} = \dfrac{\pi\sqrt{2}}{2} = \dfrac{\pi}{\sqrt{2}}$ (unstable)

$\displaystyle \prod_{n\geq1} \dfrac{2n\!-\!1}{n}.\dfrac{2n\!-\!1}{n} = 1.1.\dfrac{3}{2}.\dfrac{3}{2}.\dfrac{5}{3}.\dfrac{5}{3}.\ldots = \dfrac{\sqrt{2}}{\pi} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).(2n\!-\!1)}{\prod_{n\geq1} n.n} = \dfrac{1.1.3.3.5.5.\ldots}{1.1.2.2.3.3.\ldots} = \dfrac{2}{\pi\sqrt{2}} = \dfrac{\sqrt{2}}{\pi}$ (unstable)

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

$\displaystyle \prod_{n\geq1} \left(1\!+\!\dfrac{1}{n}\right)^n = 2.\dfrac{9}{4}.\dfrac{64}{27}.\dfrac{625}{256}.\ldots = \dfrac{1}{\sqrt{2\pi\,e}} \\ = \dfrac{\prod_{n\geq1} (n\!+\!1)^n}{\prod_{n\geq1} n^n} = \dfrac{2.9.64.625.\ldots}{1.4.27.256.\ldots} = \dfrac{\frac{1}{\sqrt{2\pi}}\,A\,e^{-\frac{7}{12}}}{A\,e^{-\frac{1}{12}}} = \dfrac{1}{\sqrt{2\pi\,e}}$ (unstable)

$\displaystyle \prod_{n\geq1} \left(1\!+\!\dfrac{1}{n}\right)^{n+1} = 4.\dfrac{27}{8}.\dfrac{256}{81}.\dfrac{3125}{1024}.\ldots = \dfrac{1}{\sqrt{2\pi\,e}} \\ = \dfrac{\prod_{n\geq1} (n\!+\!1)^{n+1}}{\prod_{n\geq1} n^{n+1}} = \dfrac{4.27.256.3125.\ldots}{1.\;\;8.\;\;81.1024.\ldots} = \dfrac{A\,e^{-\frac{7}{12}}}{\sqrt{2\pi}\,A\,e^{-\frac{1}{12}}} = \dfrac{1}{\sqrt{2\pi\,e}}$ (unstable)

Wallis products

$\displaystyle \prod_{n\geq1} \dfrac{4n^2}{4n^2\!-\!1} = \prod_{n\geq1} \dfrac{n^2}{n^2\!-\!\frac{1}{4}} = \dfrac{4}{3}.\dfrac{16}{15}.\dfrac{36}{35}.\dfrac{64}{63}.\dfrac{100}{99}.\dfrac{144}{143}.\ldots = \dfrac{\pi}{2} \\ = \dfrac{\prod_{n\geq1} 4n^2}{\prod_{n\geq1} (4n^2\!-\!1)} = \dfrac{4.16.36.64.100.144.\ldots}{3.15.35.63.\;\,99.143.\ldots} = \dfrac{(\sqrt{\pi})^2}{1.2} = \dfrac{\pi}{2}$ (convergent)

$\displaystyle \prod_{n\geq1} \dfrac{4n^2\!-\!1}{4n^2} = \prod_{n\geq1} \dfrac{n^2\!-\!\frac{1}{4}}{n^2} = \dfrac{3}{4}.\dfrac{15}{16}.\dfrac{35}{36}.\dfrac{63}{64}.\dfrac{99}{100}.\dfrac{143}{144}.\ldots = \dfrac{2}{\pi} \\ = \dfrac{\prod_{n\geq1} (4n^2\!-\!1)}{\prod_{n\geq1} 4n^2} = \dfrac{3.15.35.63.\;\,99.143.\ldots}{4.16.36.64.100.144.\ldots} = \dfrac{2}{\pi}$ (convergent)

Futuna products

$\displaystyle \prod_{n\geq1} \dfrac{2n}{2n\!-\!1}.\dfrac{2n}{2n\!+\!1} = \dfrac{2}{1}.\dfrac{2}{3}.\dfrac{4}{3}.\dfrac{4}{5}.\dfrac{6}{5}.\dfrac{6}{7}.\dfrac{8}{7}.\dfrac{8}{9}.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}}.\dfrac{\sqrt{\pi}}{\sqrt{2}} = \dfrac{\pi}{2} \\ = \dfrac{\prod_{n\geq1} 2n.2n}{\prod_{n\geq1} (2n\!-\!1).(2n\!+\!1)} = \dfrac{2.2.4.4.6.6.8.8.\ldots}{1.3.3.5.5.7.7.9.\ldots} = \dfrac{\sqrt{\pi}.\sqrt{\pi}}{\sqrt{2}\,.\,\sqrt{2}} = \dfrac{\pi}{2}$ (convergent)

$\displaystyle \prod_{n\geq1} \dfrac{2n\!-\!1}{2n}.\dfrac{2n\!+\!1}{2n} = \dfrac{1}{2}.\dfrac{3}{2}.\dfrac{3}{4}.\dfrac{5}{4}.\dfrac{5}{6}.\dfrac{7}{6}.\dfrac{7}{8}.\dfrac{9}{8}.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi}}.\dfrac{\sqrt{2}}{\sqrt{\pi}} = \dfrac{2}{\pi} \\ = \dfrac{\prod_{n\geq1} (2n\!-\!1).(2n\!+\!1)}{\prod_{n\geq1} 2n.2n} = \dfrac{1.3.3.5.5.7.7.9.\ldots}{2.2.4.4.6.6.8.8.\ldots} = \dfrac{\sqrt{2}\,.\,\sqrt{2}}{\sqrt{\pi}.\sqrt{\pi}} = \dfrac{2}{\pi}$ (convergent)

SUBSPECIES

$\displaystyle \prod_{n\geq2} n^{(-1)^n} = 2.\dfrac{1}{3}.4.\dfrac{1}{5}.6.\dfrac{1}{7}.8.\dfrac{1}{9}.\ldots = \sqrt{\dfrac{\pi}{2}} \\ = \dfrac{\prod_{n\geq2} 2n.1}{\prod_{n\geq2} 1.(2n\!-\!1)} = \dfrac{2.1.4.1.6.1.8.1.\ldots}{1.3.1.5.1.7.1.9.\ldots} = \sqrt{\dfrac{\pi}{2}}$ (unstable)

$\displaystyle \prod_{n\geq2} n^{(-1)^{n-1}} = \dfrac{1}{2}.3.\dfrac{1}{4}.5.\dfrac{1}{6}.7.\dfrac{1}{8}.9.\ldots = \sqrt{\dfrac{2}{\pi}} \\ = \dfrac{\prod_{n\geq2} 1.(2n\!-\!1)}{\prod_{n\geq2} 2n.1} = \dfrac{1.3.1.5.1.7.1.9.\ldots}{2.1.4.1.6.1.8.1.\ldots} = \sqrt{\dfrac{2}{\pi}}$ (unstable)

$\displaystyle \prod_{n\geq2} n.\dfrac{1}{n} = 2.\dfrac{1}{2}.3.\dfrac{1}{3}.4.\dfrac{1}{4}.5.\dfrac{1}{5}.\ldots = \dfrac{1}{\sqrt{6}} \\ = \dfrac{\prod_{n\geq2} n.1}{\prod_{n\geq2} 1.n} = \dfrac{2.1.3.1.4.1.5.1.\ldots}{1.2.1.3.1.4.1.5.\ldots} = \dfrac{\frac{\sqrt{\pi}}{\sqrt{6}}}{\sqrt{\pi}} = \dfrac{1}{\sqrt{6}}$ (unstable)

$\displaystyle \prod_{n\geq3} n.\dfrac{1}{n} = 3.\dfrac{1}{3}.4.\dfrac{1}{4}.5.\dfrac{1}{5}.6.\dfrac{1}{6}.\ldots = \dfrac{\sqrt{3}}{\sqrt{10}} \\ = \dfrac{\prod_{n\geq3} n.1}{\prod_{n\geq3} 1.n} = \dfrac{3.1.4.1.5.1.6.1\ldots}{1.3.1.4.1.5.1.6.\ldots} = \dfrac{\frac{\sqrt{\pi}}{4\sqrt{5}}}{\frac{\sqrt{\pi}}{2\sqrt{6}}} = \dfrac{\sqrt{3}}{\sqrt{10}}$ (unstable)

$\displaystyle \prod_{n\geq4} n.\dfrac{1}{n} = 4.\dfrac{1}{4}.5.\dfrac{1}{5}.6.\dfrac{1}{6}.7.\dfrac{1}{7}.\ldots = \dfrac{\sqrt{5}}{\sqrt{14}} \\ = \dfrac{\prod_{n\geq4} n.1}{\prod_{n\geq4} 1.n} = \dfrac{4.1.5.1.6.1.7.1.\ldots}{1.4.1.5.1.6.1.7.\ldots} = \dfrac{\frac{\sqrt{\pi}}{8\sqrt{42}}}{\frac{\sqrt{\pi}}{8\sqrt{15}}} = \dfrac{\sqrt{5}}{\sqrt{14}}$ (unstable)

$\displaystyle \prod_{n\geq5} n.\dfrac{1}{n} = 5.\dfrac{1}{5}.6.\dfrac{1}{6}.7.\dfrac{1}{7}.8.\dfrac{1}{8}.\ldots = \dfrac{\sqrt{7}}{3\sqrt{2}} \\ = \dfrac{\prod_{n\geq5} n.1}{\prod_{n\geq5} 1.n} = \dfrac{5.1.6.1.7.1.8.1.\ldots}{1.5.1.6.1.7.1.8.\ldots} = \dfrac{\frac{\sqrt{\pi}}{288\sqrt{2}}}{\frac{\sqrt{\pi}}{96\sqrt{7}}} = \dfrac{\sqrt{7}}{3\sqrt{2}}$ (unstable)

$\displaystyle \prod_{n\geq6} n.\dfrac{1}{n} = 6.\dfrac{1}{6}.7.\dfrac{1}{7}.8.\dfrac{1}{8}.9.\dfrac{1}{9}.\ldots = \dfrac{3}{\sqrt{22}} \\ = \dfrac{\prod_{n\geq6} n.1}{\prod_{n\geq6} 1.n} = \dfrac{6.1.7.1.8.1.9.1.\ldots}{1.6.1.7.1.8.1.9.\ldots} = \dfrac{\frac{\sqrt{\pi}}{384\sqrt{110}}}{\frac{\sqrt{\pi}}{1152\sqrt{5}}} = \dfrac{3}{\sqrt{22}}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!-\!\dfrac{1}{n^2}\right)^n \! = \left(\dfrac{3}{4}\right)^2\!.\left(\dfrac{8}{9}\right)^3\!.\left(\dfrac{15}{16}\right)^4\!.\left(\dfrac{24}{25}\right)^5\!.\ldots = 2$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!-\!\dfrac{1}{n^2}\right)^{n+1} \! = \left(\dfrac{3}{4}\right)^3\!.\left(\dfrac{8}{9}\right)^4\!.\left(\dfrac{15}{16}\right)^5\!.\left(\dfrac{24}{25}\right)^6\!.\ldots = 1$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!-\!\dfrac{1}{n^2}\right)^{n+2} \! = \left(\dfrac{3}{4}\right)^4\!.\left(\dfrac{8}{9}\right)^5\!.\left(\dfrac{15}{16}\right)^6\!.\left(\dfrac{24}{25}\right)^7\!.\ldots = \dfrac{1}{2}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!+\!\dfrac{1}{n}\right)^n = \dfrac{9}{4}.\dfrac{64}{27}.\dfrac{625}{256}.\dfrac{7776}{3125}.\ldots = \dfrac{\sqrt{2}}{\sqrt{\pi\,e}} \\ = \dfrac{\prod_{n\geq2} (n\!+\!1)^n}{\prod_{n\geq2} n^n} = \dfrac{9.64.625.7776.\ldots}{4.27.256.3125.\ldots} = \dfrac{\frac{\sqrt{2}}{\sqrt{\pi}}\,A\,e^{-\frac{13}{12}}}{A\,e^{-\frac{7}{12}}} = \dfrac{\sqrt{2}}{\sqrt{\pi\,e}}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!+\!\dfrac{1}{n}\right)^{n+1} = \dfrac{27}{8}.\dfrac{256}{81}.\dfrac{3125}{1024}.\dfrac{46656}{15625}.\ldots = \dfrac{1}{\sqrt{2\pi\,e}} \\ = \dfrac{\prod_{n\geq2} (n\!+\!1)^{n+1}}{\prod_{n\geq2} n^{n+1}} = \dfrac{27.256.3125.46656\ldots}{\;\;8.\;\;81.1024.15625\ldots} = \dfrac{A\,e^{-\frac{7}{12}}}{\sqrt{2\pi}\,A\,e^{-\frac{1}{12}}} = \nolinebreak \dfrac{1}{\sqrt{2\pi\,e}}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!-\!\dfrac{1}{n}\right)^n = \dfrac{1}{4}.\dfrac{8}{27}.\dfrac{81}{256}.\dfrac{1024}{3125}.\ldots = \sqrt{2\pi\,e} \\ = \dfrac{\prod_{n\geq2} (n\!-\!1)^n}{\prod_{n\geq2} n^n} = \dfrac{1.\;\;8.\;\;81.1024.\ldots}{4.27.256.3125.\ldots} = \dfrac{\sqrt{2\pi}\,A\,e^{-\frac{1}{12}}}{A\,e^{-\frac{7}{12}}} = \sqrt{2\pi\,e}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(1\!-\!\dfrac{1}{n}\right)^{n+1} = \dfrac{1}{8}.\dfrac{16}{81}.\dfrac{243}{1024}.\dfrac{4096}{15625}.\ldots = \sqrt{2\pi\,e} \\ = \dfrac{\prod_{n\geq2} (n\!-\!1)^{n+1}}{\prod_{n\geq2} n^{n+1}} = \dfrac{1.16.\;\;243.\;\;4096.\ldots}{8.81.1024.15625.\ldots} = \dfrac{2\pi\,A\,e^{-\frac{1}{12}}}{\sqrt{2\pi}\,A\,e^{-\frac{7}{12}}} = \sqrt{2\pi\,e}$ (unstable)

$\displaystyle \prod_{n\geq2} \left(\dfrac{n\!-\!1}{n\!+\!1}\right)^n = \prod_{n\geq2} \left(1\!-\!\dfrac{2}{n\!+\!1}\right)^n = \dfrac{1}{9}.\dfrac{1}{8}.\dfrac{81}{625}.\dfrac{32}{243}.\ldots = \pi\,e \\ = \dfrac{\prod_{n\geq2} (n\!-\!1)^n}{\prod_{n\geq2} (n\!+\!1)^n} = \dfrac{1.1.\;\;81.\;\;32.\ldots}{9.8.625.243.\ldots} = \dfrac{\sqrt{2\pi}\,A\,e^{-\frac{1}{12}}}{\frac{\sqrt{2}}{\sqrt{\pi}}\,A\,e^{-\frac{13}{12}}} = \pi\,e$ (unstable)

$\displaystyle \prod_{n\geq2} \left(\dfrac{n\!+\!1}{n\!-\!1}\right)^n = \prod_{n\geq2} \left(1\!+\!\dfrac{2}{n\!-\!1}\right)^n = 9.8.\dfrac{625}{81}.\dfrac{243}{32}.\ldots = \dfrac{1}{\pi\,e} \\ = \dfrac{\prod_{n\geq2} (n\!+\!1)^n}{\prod_{n\geq2} (n\!-\!1)^n} = \dfrac{9.8.625.243.\ldots}{1.1.\;\;81.\;\;32.\ldots} = \dfrac{\frac{\sqrt{2}}{\sqrt{\pi}}\,A\,e^{-\frac{13}{12}}}{\sqrt{2\pi}\,A\,e^{-\frac{1}{12}}} = \dfrac{1}{\pi\,e}$ (unstable)

INFRASPECIES

$\displaystyle \prod_{n\geq m} n^{(-1)^{n-m}} = \prod_{n\geq0} (n+m)^{(-1)^n} = m.\dfrac{1}{m\!+\!1}.(m+2).\dfrac{1}{m\!+\!3}.\ldots = \nolinebreak \sqrt{2}\,\dfrac{\Gamma(\frac{m+1}{2})}{\Gamma(\frac{m}{2})} \\ = \dfrac{\prod_{n\geq m} (2n\!-\!m).1}{\prod_{n\geq m} 1.(2n\!-\!m\!+\!1)} = \dfrac{\prod_{n\geq0} (2n\!+\!m).1}{\prod_{n\geq0} 1.(2n\!+\!m\!+\!1)} = \sqrt{2}\,\dfrac{\Gamma(\frac{m+1}{2})}{\Gamma(\frac{m}{2})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} \left(n\!-\!\frac{1}{2}\right) = \prod_{n\geq1} \left(n\!+\!m\!-\!\frac{1}{2}\right) = \dfrac{\sqrt{2\pi}}{\Gamma(m\!+\!\frac{1}{2})} = \dfrac{4^m\,m!\,\sqrt{2}}{(2m)!} \\ = \dfrac{\prod_{n\geq m+1} (2n\!-\!1)}{\prod_{n\geq m+1} 2} = \dfrac{\prod_{n\geq1} (2(n\!+\!m)\!-\!1)}{\prod_{n\geq1} 2} = \dfrac{\sqrt{2\pi}}{\Gamma(m\!+\!\frac{1}{2})} = \dfrac{4^m\,m!\,\sqrt{2}}{(2m)!}$ (unstable)

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