Simple question during a proof. Reducing a factorial...

Question

So I'm reading over a proof-review and stuck on how they managed to convert:

$\limsup \displaystyle \frac{|n!|^\frac{1}{2}}{|(n+1)!^\frac{1}{2}|} = \limsup \displaystyle\frac{1}{(n+1)^\frac{1}{2}}$

Simple factorial confusion...

Answer 1

Just remember that square roots (and taking powers in general) respect multiplication multiplicatively: $\sqrt{ab}=\sqrt{a}\sqrt{b}$, provided these are well defined. So in your problem:

$\begin{align} \frac{\sqrt{n!}}{\sqrt{(n+1)!}} &= \frac{\sqrt{n\cdot(n-1)\cdot\ldots\cdot 1}}{\sqrt{(n+1)\cdot n\cdot\ldots\cdot 1}} = \frac{\sqrt{n}\cdot\sqrt{n-1}\cdot\ldots\cdot \sqrt{1}}{\sqrt{n+1}\cdot \sqrt{n}\cdot\ldots\cdot \sqrt{1}} =\frac{1}{\sqrt{n+1}}. \end{align}$