Menu Close

Can-you-please-show-me-how-to-solve-L-lim-n-x-n-n-




Question Number 6028 by FilupSmith last updated on 10/Jun/16
Can you please show me how to solve:  L=lim_(n→∞)  (x^n /(n!))
$$\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{show}\:\mathrm{me}\:\mathrm{how}\:\mathrm{to}\:\mathrm{solve}: \\ $$$${L}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}^{{n}} }{{n}!} \\ $$
Commented by Yozzii last updated on 10/Jun/16
Assume x>0.  n!≈(√(π(2n+(1/3))))n^n e^(−n)  for large n.  ∴ For large n,   (x^n /(n!))≈(((ex)^n )/( (√(π(2n+(1/3))))n^n ))  (x^n /(n!))≈(1/( (√(π(2n+(1/3))))))(((ex)/n))^n
$${Assume}\:{x}>\mathrm{0}. \\ $$$${n}!\approx\sqrt{\pi\left(\mathrm{2}{n}+\frac{\mathrm{1}}{\mathrm{3}}\right)}{n}^{{n}} {e}^{−{n}} \:{for}\:{large}\:{n}. \\ $$$$\therefore\:{For}\:{large}\:{n},\: \\ $$$$\frac{{x}^{{n}} }{{n}!}\approx\frac{\left({ex}\right)^{{n}} }{\:\sqrt{\pi\left(\mathrm{2}{n}+\frac{\mathrm{1}}{\mathrm{3}}\right)}{n}^{{n}} } \\ $$$$\frac{{x}^{{n}} }{{n}!}\approx\frac{\mathrm{1}}{\:\sqrt{\pi\left(\mathrm{2}{n}+\frac{\mathrm{1}}{\mathrm{3}}\right)}}\left(\frac{{ex}}{{n}}\right)^{{n}} \\ $$$$ \\ $$
Commented by Yozzii last updated on 10/Jun/16
Let u=(1/n)  ∴(x^n /(n!))≈(((exu)^(1/u) )/( (√(π((2/u)+(1/3))))))    u→0 as n→∞  lim_(n→∞) (x^n /(n!))≈lim_(u→0) (((exu)^(1/u) (3u)^(0.5) )/( (√(π(6+u)))))  lim_(n→∞) (x^n /(n!))≈lim_(u→0) (((ex)^(1/u) u^((1/u)+0.5) (√3))/( (√(π(u+6)))))
$${Let}\:{u}=\frac{\mathrm{1}}{{n}} \\ $$$$\therefore\frac{{x}^{{n}} }{{n}!}\approx\frac{\left({exu}\right)^{\mathrm{1}/{u}} }{\:\sqrt{\pi\left(\frac{\mathrm{2}}{{u}}+\frac{\mathrm{1}}{\mathrm{3}}\right)}}\:\:\:\:{u}\rightarrow\mathrm{0}\:{as}\:{n}\rightarrow\infty \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{x}^{{n}} }{{n}!}\approx\underset{{u}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({exu}\right)^{\mathrm{1}/{u}} \left(\mathrm{3}{u}\right)^{\mathrm{0}.\mathrm{5}} }{\:\sqrt{\pi\left(\mathrm{6}+{u}\right)}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{x}^{{n}} }{{n}!}\approx\underset{{u}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({ex}\right)^{\mathrm{1}/{u}} {u}^{\frac{\mathrm{1}}{{u}}+\mathrm{0}.\mathrm{5}} \sqrt{\mathrm{3}}}{\:\sqrt{\pi\left({u}+\mathrm{6}\right)}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *