Question Number 155478 by alcohol last updated on 01/Oct/21
$$\underset{{k}\:=\:\mathrm{1}\:} {\overset{{n}} {\prod}}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{{k}}\right)^{{k}} \:=\:{V}_{{n}} \\ $$$${find}\:{V}_{{n}} \\ $$
Answered by mindispower last updated on 01/Oct/21
$$ \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{\left({k}+\mathrm{1}\right)^{{k}} }{{k}^{{k}} }=\Pi\frac{\left({k}+\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}^{{k}} .\left(\mathrm{1}+{k}\right)} \\ $$$$=\frac{\left(\mathrm{1}+{n}\right)^{{n}+\mathrm{1}} }{.\left({n}+\mathrm{1}\right)!}=\frac{\left(\mathrm{1}+{n}\right)^{{n}} }{{n}!} \\ $$