Friday, 15 February 2019

real analysis - Show that limlimitsntoinftyfrac(n!)1/nn=frac1e





Show that lim




What I did is to let U_n = \dfrac{(n!)^{\frac{1}{n}}}{n} and U_{n+1} = \dfrac{(n+1)!^{\frac{1}{n+1}}}{n+1}. Then



\frac{ U_{n+1} }{U_n } = \frac{\frac{(n+1)!^{\frac{1}{n+1}}}{n+1}}{\frac{(n!)^{\frac{1}{n}}}{n}}




Next I just got stuck. Am I on the right track, or am I wrong doing this type of sequence?


Answer



Let v_n = \frac{n!}{n^n } then \frac{v_{n+1}}{v_n } =\frac{(n+1)! }{(n+1)^{n+1}} \cdot \frac{n^n }{n!} =\frac{n^n}{(n+1)^n }=\frac{1}{\left(1+\frac{1}{n}\right)^n}\to \frac{1}{e} hence \frac{\sqrt[n]{n!} }{n} =\sqrt[n]{v_n} \to\frac{1}{e} .


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...