Divergent Products of Negative Integers

HYPERSPECIES

$\displaystyle \prod_{\substack{n\geq1 \\ j\text{-spacing} \\ k\text{ shifts}}} (-1) = \underbrace{1.\ldots.1}_{k\text{ shifts}}.(-1).\underbrace{1.\ldots.1}_{j\text{-spacing}}.(-1).\underbrace{1.\ldots.1}_{j\text{-spacing}}.\ldots = (-1)^{\mp\left(\frac{1}{2}+\frac{k-j}{j+1}\right)}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1 \\ \text{no spacing} \\ k\text{ shifts}}} (-1) = \underbrace{1.\ldots.1}_{k\text{ shifts}}.(-1).(-1).(-1).\ldots = (-1)^{\mp\left(\frac{1}{2}+k\right)} = \mp i(-1)^k$ (unstable)

$\displaystyle \prod_{\substack{n\geq1 \\ 1\text{-spacing} \\ k\text{ shifts}}} (-1) = \underbrace{1.\ldots.1}_{k\text{ shifts}}.(-1).1.(-1).1.(-1).1.\ldots = (-1)^{\mp\frac{k}{2}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1 \\ 2\text{-spacing} \\ k\text{ shifts}}} (-1) = \underbrace{1.\ldots.1}_{k\text{ shifts}}.(-1).1.1.(-1).1.1.(-1).1.1.\ldots = (-1)^{\pm\left(\frac{1}{6}-\frac{k}{3}\right)}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1 \\ 3\text{-spacing} \\ k\text{ shifts}}} (-1) = \underbrace{1.\ldots.1}_{k\text{ shifts}}.(-1).1.1.1.(-1).1.1.1.(-1).1.1.1.\ldots = (-1)^{\mp\frac{k-1}{4}}$ (unstable)

SUPERSPECIES

$\displaystyle \prod_{n\geq0} (-1)^n = \prod_{n\geq1} 1.(-1) = 1.(-1).1.(-1).\ldots = (-1)^{\mp\frac{1}{2}} = i^{\mp1} = \nolinebreak \mp i \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1)^n = \prod_{n\geq1} (-1).1 = (-1).1.(-1).1.\ldots = (-1)^0 = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.1 = (-1).1.1.(-1).1.1.(-1).1.1.\ldots \\ = (-1)^{\pm\frac{1}{6}} = e^{\pm\frac{i\pi}{6}} = \frac{\sqrt{3}}{2} \pm \frac{i}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).1 = 1.(-1).1.1.(-1).1.1.(-1).1.\ldots \\ = (-1)^{\mp\frac{1}{6}} = e^{\mp\frac{i\pi}{6}} = \frac{\sqrt{3}}{2} \mp \nolinebreak \frac{i}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.1.(-1) = 1.1.(-1).1.1.(-1).1.1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{2}} = e^{\mp\frac{i\pi}{2}} = \mp i \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).(-1).1 = (-1).(-1).1.(-1).(-1).1.(-1).(-1).1.\ldots \\ = (-1)^0 = e^0 = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.(-1) = (-1).1.(-1).(-1).1.(-1).(-1).1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{3}} = e^{\mp\frac{i\pi}{3}} = \frac{1}{2} \mp \frac{i\sqrt{3}}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).(-1) = 1.(-1).(-1).1.(-1).(-1).1.(-1).(-1).\ldots \\ = (-1)^{\mp\frac{2}{3}} = e^{\mp\frac{2i\pi}{3}} = -\frac{1}{2} \mp \frac{i\sqrt{3}}{2} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.1.1 = (-1).1.1.1.(-1).1.1.1.(-1).1.1.1.\ldots \\ = (-1)^{\pm\frac{1}{4}} = e^{\pm\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \pm \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).1.1 = 1.(-1).1.1.1.(-1).1.1.1.(-1).1.1.\ldots \\ = (-1)^0 = e^0 = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.1.(-1).1 = 1.1.(-1).1.1.1.(-1).1.1.1.(-1).1.\ldots \\ = (-1)^{\mp\frac{1}{4}} = e^{\mp\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.1.1.(-1) = 1.1.1.(-1).1.1.1.(-1).1.1.1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{2}} = e^{\mp\frac{i\pi}{2}} = \mp i \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).(-1).1.1 = (-1).(-1).1.1.(-1).(-1).1.1.(-1).(-1).1.1.\ldots \\ = (-1)^{\pm\frac{1}{4}} = e^{\pm\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \pm \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.(-1).1 = \prod_{n\geq1} (-1).1 = (-1).1.(-1).1.(-1).1.(-1).1.(-1).1.(-1).\ldots \\ = (-1)^0 = e^0 = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.1.(-1) = (-1).1.1.(-1).(-1).1.1.(-1).(-1).1.1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{4}} = e^{\mp\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).(-1).1 = 1.(-1).(-1).1.1.(-1).(-1).1.1.(-1).(-1).1.\ldots \\ = (-1)^{\mp\frac{1}{4}} = e^{\mp\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).1.(-1) = \prod_{n\geq1} 1.(-1) = 1.(-1).1.(-1).1.(-1).1.(-1).1.(-1).1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{2}} = e^{\mp\frac{i\pi}{2}} = \mp i \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.1.(-1).(-1) = 1.1.(-1).(-1).1.1.(-1).(-1).1.1.(-1).(-1).\ldots \\ = (-1)^{\mp\frac{3}{4}} = e^{\mp\frac{3i\pi}{4}} = -\frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).(-1).(-1).1 = (-1).(-1).(-1).1.(-1).(-1).(-1).1.(-1).(-1).(-1).1.\ldots \\ = (-1)^0 = e^0 = 1 \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).(-1).1.(-1) = (-1).(-1).1.(-1).(-1).(-1).1.(-1).(-1).(-1).1.(-1).\ldots \\ = (-1)^{\mp\frac{1}{4}} = e^{\mp\frac{i\pi}{4}} = \frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} (-1).1.(-1).(-1) = (-1).1.(-1).(-1).(-1).1.(-1).(-1).(-1).1.(-1).(-1).\ldots \\ = (-1)^{\mp\frac{1}{2}} = e^{\mp\frac{i\pi}{2}} = \mp i \qquad \textit{(unstable)}$

$\displaystyle \prod_{n\geq1} 1.(-1).(-1).(-1) = 1.(-1).(-1).(-1).1.(-1).(-1).(-1).1.(-1).(-1).(-1).\ldots \\ = (-1)^{\mp\frac{3}{4}} = e^{\mp\frac{3i\pi}{4}} = -\frac{1}{\sqrt{2}} \mp \frac{i}{\sqrt{2}} \qquad \textit{(unstable)}$

SPECIES

$\displaystyle \prod_{n\geq1} (-1) = (-1).(-1).(-1).(-1).(-1).(-1).\ldots = (-1)^{\mp\frac{1}{2}} = i^{\mp1} = \nolinebreak \mp i \qquad \textit{(unstable)}$