Divergent Products of Integers

HYPERVARIETIES

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 2 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.2.2.2.2.2.2.\ldots = \dfrac{1}{2^{k+\frac{1}{2}}} = \underbrace{\dfrac{1}{\sqrt{2}}}_{\text{seed}} \underbrace{\dfrac{1}{2^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 2.1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.2.1.2.1.2.1.\ldots = \dfrac{1}{2^\frac{k}{2}} = \underbrace{\dfrac{1}{\sqrt{2}}}_{\text{seed}} \underbrace{\dfrac{1}{2^\frac{k-1}{2}}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.2.1.2.1.2.\ldots = \dfrac{1}{2^\frac{k+1}{2}} = \underbrace{\dfrac{1}{\sqrt{2}}}_{\text{seed}} \underbrace{\dfrac{1}{2^\frac{k}{2}}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 3 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.3.3.3.3.3.\ldots = \dfrac{1}{3^{k+\frac{1}{2}}} = \underbrace{\dfrac{1}{\sqrt{3}}}_{\text{seed}} \underbrace{\dfrac{1}{3^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 3.1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.1.3.1.3.1.\ldots = \dfrac{1}{3^\frac{k}{2}} = \underbrace{\dfrac{1}{\sqrt{3}}}_{\text{seed}} \underbrace{\dfrac{1}{3^\frac{k-1}{2}}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.3 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.3.1.3.1.3.\ldots = \dfrac{1}{3^\frac{k+1}{2}} = \underbrace{\dfrac{1}{\sqrt{3}}}_{\text{seed}} \underbrace{\dfrac{1}{3^\frac{k}{2}}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 2n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.2.4.6.8.10.12.\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{2}} \; \dfrac{1}{2^k} = \underbrace{\sqrt{\pi}}_{\text{seed}} \, \underbrace{\dfrac{1}{2^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 2n.1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.2.1.4.1.6.1.\ldots = \underbrace{\sqrt{\pi}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.2n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.2.1.4.1.6.\ldots = \underbrace{\sqrt{\pi}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 3n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.6.9.12.15.18.\ldots = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{3}}}_{\text{seed}} \underbrace{\dfrac{1}{3^k}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 3n.1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.1.6.1.9.1.\ldots = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{3}}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.3n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.3.1.6.1.9.\ldots = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{3}}}_{\text{seed}} \qquad \textit{(semi-stable)}$

SUPERVARIETIES

$\displaystyle \prod_{n\geq1} (n+1)^\frac{1-(-1)^n}{2} = \prod_{n\geq1} 2n.1 = 2.1.4.1.6.1.8.1.\ldots = \sqrt{\pi} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} n^\frac{1+(-1)^n}{2} = \prod_{n\geq1} 1.2n = 1.2.1.4.1.6.1.8.1.\ldots = \sqrt{\pi} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} 2n.2 = 2.2.4.2.6.2.8.2.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 2.2n = 2.2.2.4.2.6.2.8.\ldots = \sqrt{\pi} \qquad \textit{(unstable)}$

VARIETIES

$\displaystyle \prod_{n\geq1} 2n = 2.4.6.8.10.12.\ldots = \dfrac{1}{\sqrt{2}} \, \sqrt{2\pi} = \sqrt{\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 4n^2 = 4.16.36.64.100.144.\ldots = \dfrac{(\sqrt{2\pi})^2}{2} = (\sqrt{\pi})^2 = \pi \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 3n = 3.6.9.12.15.18.\ldots = \dfrac{1}{\sqrt{3}} \, \sqrt{2\pi} = \dfrac{\sqrt{2\pi}}{\sqrt{3}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 9n^2 = 9.36.81.144.225.324.\ldots = \dfrac{(\sqrt{2\pi})^2}{3} = \left(\dfrac{\sqrt{2\pi}}{\sqrt{3}}\right)^2 = \nolinebreak \dfrac{2\pi}{3} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \left(n(n\!+\!1)\right)^\frac{1}{2} \left(\dfrac{n}{n\!+\!1}\right)^\frac{(-1)^n}{2} = \prod_{n\geq2} 2\left\lfloor\dfrac{n}{2}\right\rfloor = \prod_{n\geq1} 2\left\lfloor\dfrac{n\!+\!1}{2}\right\rfloor = \prod_{n\geq1} 2\left\lceil\dfrac{n}{2}\right\rceil \\ = \prod_{n\geq1} 2n.2n = 2.2.4.4.6.6.8.8.\ldots = \dfrac{\pi\sqrt{2}}{\sqrt{2}} = \pi \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 3n.3n = 3.3.6.6.9.9.12.12.\ldots = \dfrac{\pi\sqrt{2}}{\sqrt{3}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 4n.4n = 4.4.8.8.12.12.16.16.\ldots = \dfrac{\pi\sqrt{2}}{\sqrt{4}} = \dfrac{\pi}{\sqrt{2}} \qquad \textit{(unstable)}$

SUBVARIETIES

$\displaystyle \prod_{n\geq2} 2n = \prod_{n\geq1} 2(n\!+\!1) = 4.6.8.10.12.14.\ldots = \sqrt{\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} 2n = \prod_{n\geq1} 2(n\!+\!2) = 6.8.10.12.14.16.\ldots = \dfrac{\sqrt{\pi}}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} 2n = \prod_{n\geq1} 2(n\!+\!3) = 8.10.12.14.16.18.\ldots = \dfrac{\sqrt{\pi}}{6} \quad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} 3n = \prod_{n\geq1} 3(n\!+\!1) = 6.9.12.15.18.21.\ldots = \dfrac{\sqrt{2\pi}}{\sqrt{3}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} 3n = \prod_{n\geq1} 3(n+2) = 9.12.15.18.21.24.\ldots = \dfrac{\sqrt{2\pi}}{2\sqrt{3}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} 3n = \prod_{n\geq1} 3(n\!+\!3) = 12.15.18.21.24.27.\ldots = \dfrac{\sqrt{2\pi}}{6\sqrt{3}} \quad \textit{(unstable)}$

INFRAVARIETIES

$\displaystyle \prod_{n\geq m+1} 2 = 2.2.2.2.2.2.\ldots = \underbrace{\dfrac{1}{\sqrt{2}}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} 2.2 = \prod_{n\geq m+1} 2 = 2.2.2.2.2.2.\ldots = \underbrace{\dfrac{1}{\sqrt{2}}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} 2n = \prod_{n\geq1} 2(n+m) = \dfrac{\sqrt{\pi}}{\Gamma(m\!+\!1)} = \underbrace{\sqrt{\pi}}_{\text{seed}} \, \underbrace{\dfrac{1}{m!}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} 3 = 3.3.3.3.3.3.\ldots = \underbrace{\dfrac{1}{\sqrt{3}}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} 3.3 = \prod_{n\geq m+1} 3 = 3.3.3.3.3.3.\ldots = \underbrace{\dfrac{1}{\sqrt{3}}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} 3n = \prod_{n\geq1} 3(n+m) = \dfrac{\sqrt{2\pi}}{\sqrt{3}\,\Gamma(m\!+\!1)} = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{3}}}_{\text{seed}} \, \underbrace{\dfrac{1}{m!}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(unstable)}$

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