Divergent Products of Integers

HYPERSPECIES

$\displaystyle \prod_{\substack{n\geq m+1\\k \text{ shifts}}} n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.(m+1).(m+2).(m+3).\,\ldots = \underbrace{\sqrt{2\pi}}_{\text{seed}} \underbrace{\dfrac{1}{m!}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.2.3.4.5.6.\,\ldots = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{1}\;\Gamma(\frac{1}{1})}}_{\text{seed}} \underbrace{\dfrac{1}{1^k}}_{\substack{\text{shift} \\ \text{factor}}} = \underbrace{\sqrt{2\pi}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} n.1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.1.2.1.3.1.\ldots = 2^\frac{k}{2} \, \sqrt{\pi} = \underbrace{\sqrt{2\pi}}_{\text{seed}} \, \underbrace{2^\frac{k-1}{2}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} 1.n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.1.1.2.1.3.\ldots = 2^\frac{k+1}{2} \, \sqrt{\pi} = \underbrace{\sqrt{2\pi}}_{\text{seed}} \, \underbrace{2^\frac{k}{2}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} (2n-1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.3.5.7.9.11.\ldots = \underbrace{\dfrac{\sqrt{2\pi}}{\sqrt{2}\;\Gamma(\frac{1}{2})}}_{\text{seed}} \underbrace{\dfrac{1}{2^k}}_{\substack{\text{shift} \\ \text{factor}}} = \nolinebreak \underbrace{\dfrac{1}{2^k}}_{\substack{\text{shift} \\ \text{factor}}} \quad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} (2n-1).1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.1.3.1.5.1.7.1.\ldots = \underbrace{\sqrt{2}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} 1.(2n-1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.1.1.3.1.5.1.7.\ldots = \underbrace{\sqrt{2}}_{\text{seed}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} (2n+1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.5.7.9.11.13.\ldots = \underbrace{\dfrac{1}{2^{k-1}}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} (2n+1).1 = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.3.1.5.1.7.1.9.1.\ldots = \underbrace{\sqrt{2}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1 \\k \text{ shifts}}} 1.(2n+1) = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.3.1.5.1.7.1.9.\ldots = \underbrace{\sqrt{2}}_{\text{seed}} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} n^n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.1.2^2.3^3.4^4.5^5.6^6.\ldots = \underbrace{A \, e^{-\frac{1}{12}}}_{\text{seed}} \underbrace{e^\frac{k^2}{2}}_{\substack{\text{shift} \\ \text{factor}}} \qquad \textit{(unstable)}$

SUPERSPECIES

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

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

$\displaystyle \prod_{n\geq1} n.1 = 1.1.2.1.3.1.4.1.\ldots = \sqrt{\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.n = 1.1.1.2.1.3.1.4.\ldots = \sqrt{2\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n.2 = 1.2.2.2.3.2.4.2.\ldots = \sqrt{\dfrac{\pi}{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 2.n = 2.1.2.2.2.3.2.4.\ldots = \sqrt{2\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (n+1)^\frac{1+(-1)^n}{2} = \prod_{n\geq1} (2n+1).1 = 3.1.5.1.7.1.9.\ldots = \sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (n+1)^\frac{1+(-1)^n}{2} = \prod_{n\geq1} 1.(2n+1) = 1.3.1.5.1.7.1.9.\ldots = \sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n^\frac{1-(-1)^n}{2} = \prod_{n\geq1} (2n-1).1 = 1.1.3.1.5.1.7.1.\ldots = \sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(2n\!-\!1) = 1.1.1.3.1.5.1.7.\ldots = \sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(4n\!-\!1) = 1.3.1.7.1.11.1.15.\ldots = \dfrac{\Gamma(\frac{1}{4})}{2^\frac{1}{4}\,\sqrt{6\pi}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (4n\!-\!3).1 = 1.1.5.1.9.1.13.1.\ldots = \dfrac{2^\frac{1}{4}\,\sqrt{6\pi}}{\Gamma(\frac{1}{4})} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq0} n^n = 1 \,.\, 1 \,.\, 2^2 \,.\, 3^3 \,.\, 4^4 \,.\, 5^5 \,.\, 6^6 \,.\, \ldots = A \, e^{\frac{5}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq0} n! = \prod_{n\geq1} \Gamma(n) = 1 \,.\, 1 \,.\, 2 \,.\, 6 \,.\, 24 \,.\, 120 \,.\, 720 \,.\, \ldots = \dfrac{1}{(2\pi)^{\frac{1}{4}} \, A \, e^{-\frac{1}{12}}} \qquad \textit{(unstable)}$

SPECIES

$\displaystyle \prod_{n\geq0} n = 0.1.2.3.4.5.6.7.\ldots = 0 \qquad \textit{(convergent)}$

$\displaystyle \prod_{n\geq1} n = \prod_{n\geq1} (2n-1).2n = 1.2.3.4.5.6.7.8.\ldots = \sqrt{2\pi} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} (2n-1) = \prod_{n\geq1} (4n-3).(4n-1) = 1.3.5.7.9.11.13.15.\ldots = \nolinebreak 1 \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq1} n^2 = 1.4.9.16.25.36.49.64.\ldots = 2\pi \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n.n = 1.1.2.2.3.3.4.4.\ldots = \pi\sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \left(2\left\lfloor\dfrac{n\!-\!1}{2}\right\rfloor\!+\!1\right) = \prod_{n\geq0} \left(2\left\lfloor\dfrac{n}{2}\right\rfloor\!+\!1\right) = 1.1.3.3.5.5.7.7.\ldots = \nolinebreak 2 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \left(n(n\!+\!1)\right)^\frac{1}{2} \left(\dfrac{n\!+\!1}{n}\right)^\frac{(-1)^n}{2} = \prod_{n\geq1} \left(2\left\lfloor\dfrac{n}{2}\right\rfloor\!+\!1\right) \\ = \prod_{n\geq1} (2n-1).(2n+1) = 1.3.3.5.5.7.7.9.9.\ldots = 2 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \dfrac{n(n\!+\!1)}{2} = 1.3.6.10.15.21.28.36.\ldots = 2\pi\sqrt{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n(2n\!-\!1) = 1.6.15.28.45.66.91.120.\ldots = \sqrt{2\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (2n\!-\!1)^2 = 1.9.25.49.81.121.169.225.\ldots = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (4n^2\!-\!1) = 3.15.35.63.99.143.195.255.\ldots = 2 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} e^n = e \,.\, e^2 \,.\, e^3 \,.\, e^4 \,.\, e^5 \,.\, e^6 \,.\, \ldots = e^{-\frac{1}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n^n = 1 \,.\, 2^2 \,.\, 3^3 \,.\, 4^4 \,.\, 5^5 \,.\, 6^6 \,.\, \ldots = A \, e^{-\frac{1}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} \left(\frac{n}{e}\right)^n = \frac{1}{e} \,. \left(\frac{2}{e}\right)^2 . \left(\frac{3}{e}\right)^3 . \left(\frac{4}{e}\right)^4 . \left(\frac{5}{e}\right)^5 . \left(\frac{6}{e}\right)^6 .\, \ldots = A \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} n! = \prod_{n\geq2} \Gamma(n) = 1 \,.\, 2 \,.\, 6 \,.\, 24 \,.\, 120 \,.\, 720 \,.\, \ldots = \dfrac{(2\pi)^{\frac{1}{4}}}{A \, e^{-\frac{1}{12}}} \qquad \textit{(unstable)}$

SUBSPECIES

$\displaystyle \prod_{n\geq2} n = \prod_{n\geq1} (n\!+\!1) = 2.3.4.5.6.7.\ldots = \sqrt{2\pi} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq3} n = \prod_{n\geq1} (n\!+\!2) = 3.4.5.6.7.8.\ldots = \dfrac{\sqrt{2\pi}}{2} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq4} n = \prod_{n\geq1} (n\!+\!3) = 4.5.6.7.8.9.\ldots = \dfrac{\sqrt{2\pi}}{6} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq5} n = \prod_{n\geq1} (n\!+\!4) = 5.6.7.8.9.10.\ldots = \dfrac{\sqrt{2\pi}}{24} \qquad \textit{(semi-stable)}$

$\displaystyle \prod_{n\geq2} (2n\!-\!1) = \prod_{n\geq1} (2n\!+\!1) = 3.5.7.9.11.13.\ldots = 2 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} (2n-1) = \prod_{n\geq1} (2n+3) = 5.7.9.11.13.15.\ldots = \dfrac{4}{3} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} (2n-1) = \prod_{n\geq1} (2n+5) = 7.9.11.13.15.17.\ldots = \dfrac{8}{15} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq5} (2n-1) = \prod_{n\geq1} (2n+7) = 9.11.13.15.17.19.\ldots = \dfrac{16}{105} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} n.1 = 2.1.3.1.4.1.5.1.\ldots = \dfrac{\sqrt{\pi}}{\sqrt{6}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} n.1 = 3.1.4.1.5.1.6.1.\ldots = \dfrac{\sqrt{\pi}}{4\sqrt{5}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} n.1 = 4.1.5.1.6.1.7.1.\ldots = \dfrac{\sqrt{\pi}}{8\sqrt{42}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq5} n.1 = 5.1.6.1.7.1.8.1.\ldots = \dfrac{\sqrt{\pi}}{288\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq6} n.1 = 6.1.7.1.8.1.9.1.\ldots = \dfrac{\sqrt{\pi}}{384\sqrt{110}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} 1.n = 1.2.1.3.1.4.1.5.1.\ldots = \sqrt{\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} 1.n = 1.3.1.4.1.5.1.6.1.\ldots = \dfrac{\sqrt{\pi}}{2\sqrt{6}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} 1.n = 1.4.1.5.1.6.1.7.1.\ldots = \dfrac{\sqrt{\pi}}{8\sqrt{15}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq5} 1.n = 1.5.1.6.1.7.1.8.1.\ldots = \dfrac{\sqrt{\pi}}{96\sqrt{7}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} n^2 = 4.9.16.25.36.49.\ldots = 2\pi \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} n.n = 2.2.3.3.4.4.5.5.\ldots = \dfrac{\pi}{\sqrt{6}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq3} n.n = 3.3.4.4.5.5.6.6.\ldots = \dfrac{\pi}{8\sqrt{30}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} n.n = 4.4.5.5.6.6.7.7.\ldots = \dfrac{\pi}{192\sqrt{70}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq5} n.n = 5.5.6.6.7.7.8.8.\ldots = \dfrac{\pi}{27648\sqrt{14}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} \left(2\left\lfloor\dfrac{n}{2}\right\rfloor\!+\!1\right) = \prod_{n\geq1} (2n+1).(2n+1) = 3.3.5.5.7.7.9.9.\ldots = \nolinebreak 2 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq4} \left(2\left\lfloor\dfrac{n}{2}\right\rfloor\!+\!1\right) = \prod_{n\geq2} (2n+1).(2n+1) = 5.5.7.7.9.9.11.11.\ldots = \nolinebreak \dfrac{2}{9} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq6} \left(2\left\lfloor\dfrac{n}{2}\right\rfloor\!+\!1\right) = \prod_{n\geq3} (2n+1).(2n+1) = 7.7.9.9.11.11.13.13.\ldots = \nolinebreak \dfrac{2}{225} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} (n-1)^n = 1\,.\,2^3\,.\,3^4\,.\,4^5\,.\,5^6\,.\,6^7\,.\ldots = A \, e^{-\frac{1}{12}} \, \sqrt{2\pi} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} n^n = 2^2\,.\,3^3\,.\,4^4\,.\,5^5\,.\,6^6\,.\,7^7\,.\ldots = A \, e^{-\frac{7}{12}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} (n+1)^n = 3^2\,.\,4^3\,.\,5^4\,.\,6^5\,.\,7^6\,.\,8^7\,.\ldots = A \, e^{-\frac{13}{12}} \, \dfrac{\sqrt{2}}{\sqrt{\pi}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq2} n^{2n} = \prod_{n\geq2} (n^n)^2 = \prod_{n\geq2} (n^2)^n = 2^4\,.\,3^6\,.\,4^8\,.\,5^{10}\,.\,6^{12}\,.\,7^{14}\,.\ldots = \nolinebreak A^2 \, e^{-\frac{7}{6}} \qquad \textit{(unstable)}$

INFRASPECIES

$\displaystyle \prod_{n\geq m+1} 0 = \prod_{n\geq1} 0 = 0.0.0.0.0.0.\ldots = 0 \qquad \textit{(convergent)}$

$\displaystyle \prod_{n\geq m+1} 1 = \prod_{n\geq1} 1 = 1.1.1.1.1.1.\ldots = 1 \qquad \textit{(convergent)}$

$\displaystyle \prod_{n\geq m} n = \prod_{n\geq0} (n\!+\!m) = \dfrac{\sqrt{2\pi}}{\Gamma(m)} = \underbrace{\sqrt{2\pi}}_{\text{seed}} \underbrace{\dfrac{1}{(m\!-\!1)!}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(semi-stable)}$

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

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

$\displaystyle \prod_{n\geq m+1} n^m = \prod_{n\geq1} (n\!+\!m)^m = \dfrac{(2\pi)^{\frac{m}{2}}}{(\Gamma(m\!+\!1))^m} = \dfrac{(2\pi)^{\frac{m}{2}}}{(m!)^m} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} n^n = \prod_{n\geq1} (n\!+\!m)^{n+m} = \underbrace{A\,e^{-\frac{1}{12}}}_{\substack{\text{seed}}} \underbrace{e^{-\frac{m}{2}}}_{\substack{\text{truncation} \\ \text{factor}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} n^{n-m} = \prod_{n\geq1} (n+m)^n = \dfrac{(\Gamma(m\!+\!1))^m}{(2\pi)^{\frac{m}{2}}} \; A \, e^{-\frac{1}{12}-\frac{m}{2}} = \dfrac{(m!)^m}{(2\pi)^{\frac{m}{2}}} \; A \, e^{-\frac{1}{12}-\frac{m}{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} n! = \prod_{n\geq1} (n\!+\!m)! = \dfrac{(2\pi e)^{\frac{m}{2}}}{(m!)^{m+1}} \; \dfrac{(2\pi)^\frac{1}{4}}{A\,e^{-\frac{1}{12}}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq m+1} \Gamma(n\!+\!1) = \prod_{n\geq1} \Gamma(n\!+\!m\!+\!1) = \dfrac{(2\pi e)^{\frac{m}{2}}}{(\Gamma(m\!+\!1))^{m+1}} \; \dfrac{(2\pi)^\frac{1}{4}}{A\,e^{-\frac{1}{12}}} \qquad \textit{(unstable)}$

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