Divergent Products of Real Numbers

HYPERSPECIES

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} e^n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.e.e^2.e^3.e^4.e^5.e^6.\ldots = e^{-\frac{1}{12}+\frac{k^2}{2}}$ (unstable)

$\displaystyle \prod_{\substack{n\geq1\\k \text{ shifts}}} \left(\dfrac{n}{e}\right)^n = \underbrace{1.\,\ldots\,.1}_{k \text{ shifts}}.\dfrac{1}{e}\,.\left(\dfrac{2}{e}\right)^2.\left(\dfrac{3}{e}\right)^3.\left(\dfrac{4}{e}\right)^4.\,\ldots = A$ (semi-stable)

SPECIES

$\displaystyle \prod_{n\geq0} \sqrt{n} = 0\,.\,1\,.\,\sqrt{2}\,.\,\sqrt{3}\,.\,2\,.\,\sqrt{5}\,.\,\sqrt{6}\,.\,\ldots = 0$ (convergent)

$\displaystyle \prod_{n\geq1} \sqrt{n} = 1\,.\,\sqrt{2}\,.\,\sqrt{3}\,.\,2\,.\,\sqrt{5}\,.\,\sqrt{6}\,.\,\ldots = (2\pi)^\frac{1}{4}$ (semi-stable)

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

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

SUBSPECIES

$\displaystyle \prod_{n\geq1} \dfrac{n! \; n^n}{\sqrt{n}} = 1$ (unstable)

$\displaystyle \prod_{n\geq1} \sqrt{n} \; \Gamma(n) \; n^n = 1$ (unstable)

Infinite products related to the Stirling’s approximation for factorials

$\displaystyle \prod_{n\geq0} \left(\dfrac{n}{e}\right)^n = 1\,.\,\dfrac{1}{e}\,.\left(\dfrac{2}{e}\right)^2.\left(\dfrac{3}{e}\right)^3.\left(\dfrac{4}{e}\right)^4.\,\ldots = A$ (semi-stable)

$\displaystyle \prod_{n\geq1} \left(\dfrac{n}{e}\right)^n = \dfrac{1}{e}\,.\left(\dfrac{2}{e}\right)^2.\left(\dfrac{3}{e}\right)^3.\left(\dfrac{4}{e}\right)^4.\,\ldots = A$ (semi-stable)

$\displaystyle \prod_{n\geq1} \sqrt{2\pi n} \left(\dfrac{n}{e}\right)^n = \dfrac{\sqrt{2\pi}}{e}.\dfrac{8\sqrt{2\pi}}{e^2}.\dfrac{27\sqrt{6\pi}}{e^3}.\dfrac{512\sqrt{\pi}}{e^4}.\,\ldots = A$ (unstable)

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