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Question Number 3035 by 123456 last updated on 03/Dec/15

ω(z)=lim_(n→∞) ((n^z [z+(z+1)+...+(z+n)])/(z(z+1)(z+2)...(z+n))) ω(1)=?

$$\omega\left({z}\right)=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}^{{z}} \left[{z}+\left({z}+\mathrm{1}\right)+...+\left({z}+{n}\right)\right]}{{z}\left({z}+\mathrm{1}\right)\left({z}+\mathrm{2}\right)...\left({z}+{n}\right)} \\ $$$$\omega\left(\mathrm{1}\right)=? \\ $$

Commented by prakash jain last updated on 04/Dec/15

w(1)=lim_(n→∞) ((nΣ_(k=1) ^(1+n) k)/(1∙2∙3∙...∙(1+n)))=((n(n+1)(n+2))/(2(n+1)!)) = lim_(n→∞) (((n+1))/(2(n−1)!)) =lim_(n→∞) (((n−1))/(2(n−1)!))+(1/((n−1)!))= lim_(n→∞) (1/(2(n−2)!))+(1/((n−1)!))

$${w}\left(\mathrm{1}\right)=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}\underset{{k}=\mathrm{1}} {\overset{\mathrm{1}+{n}} {\sum}}{k}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}\centerdot...\centerdot\left(\mathrm{1}+{n}\right)}=\frac{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}{\mathrm{2}\left({n}+\mathrm{1}\right)!} \\ $$$$=\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\left({n}+\mathrm{1}\right)}{\mathrm{2}\left({n}−\mathrm{1}\right)!}\:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}−\mathrm{1}\right)}{\mathrm{2}\left({n}−\mathrm{1}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}= \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{2}\left({n}−\mathrm{2}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!} \\ $$

Answered by Filup last updated on 04/Dec/15

ω(1)=lim_(n→∞) ((n^1 [1+2+...+(n+1)])/(((((n+1)!)/((z−1)!))))) =((n^1 [Σ_(i=1) ^(n+1) i])/(((((n+1)!)/((z−1)!))))) =((n^1 [(1/2)(n+1)^2 ])/(((((n+1)!)/((z−1)!))))) =((n^1 (1/2)(n+1)^2 0!)/((n+1)!)) =((n^1 (1/2)(n+1)^2 )/((n+1)!)) =((n^1 (1/2)(n+1)^2 )/((n)!(n+1))) =((n^1 (n+1))/(2n!)) =((n^1 (n+1))/(2(n−1)!n)) ω(1)=lim_(n→∞) (((n+1))/(2(n−1)!)) ω(1)=(1/2) lim_(n→∞) (n/((n−1)!))+(1/((n−1)!)) lim_(n→∞) (n/((n−1)!)) =lim_(n→∞) (((n−1)+1)/((n−1)(n−2)!)) =lim_(n→∞) (((n−1))/((n−1)(n−2)!))+(1/((n−1)(n−2)!)) =lim_(n→∞) (1/((n−2)!))+(1/((n−1)(n−2)!)) =0 lim_(n→∞) (1/((n−1)!)) = 0 ω(1)=(1/2)(0+0) ∴ ω(1)=0

$$\omega\left(\mathrm{1}\right)=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}^{\mathrm{1}} \left[\mathrm{1}+\mathrm{2}+...+\left({n}+\mathrm{1}\right)\right]}{\left(\frac{\left({n}+\mathrm{1}\right)!}{\left({z}−\mathrm{1}\right)!}\right)} \\ $$$$=\frac{{n}^{\mathrm{1}} \left[\underset{{i}=\mathrm{1}} {\overset{{n}+\mathrm{1}} {\sum}}{i}\right]}{\left(\frac{\left({n}+\mathrm{1}\right)!}{\left({z}−\mathrm{1}\right)!}\right)} \\ $$$$=\frac{{n}^{\mathrm{1}} \left[\frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)^{\mathrm{2}} \right]}{\left(\frac{\left({n}+\mathrm{1}\right)!}{\left({z}−\mathrm{1}\right)!}\right)} \\ $$$$=\frac{{n}^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)^{\mathrm{2}} \mathrm{0}!}{\left({n}+\mathrm{1}\right)!} \\ $$$$=\frac{{n}^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }{\left({n}+\mathrm{1}\right)!} \\ $$$$=\frac{{n}^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }{\left({n}\right)!\left({n}+\mathrm{1}\right)} \\ $$$$=\frac{{n}^{\mathrm{1}} \left({n}+\mathrm{1}\right)}{\mathrm{2}{n}!} \\ $$$$=\frac{{n}^{\mathrm{1}} \left({n}+\mathrm{1}\right)}{\mathrm{2}\left({n}−\mathrm{1}\right)!{n}} \\ $$$$\omega\left(\mathrm{1}\right)=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}+\mathrm{1}\right)}{\mathrm{2}\left({n}−\mathrm{1}\right)!} \\ $$$$\omega\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{n}}{\left({n}−\mathrm{1}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!} \\ $$$$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{n}}{\left({n}−\mathrm{1}\right)!}\:=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}−\mathrm{1}\right)+\mathrm{1}}{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)!} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left({n}−\mathrm{1}\right)}{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)!} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\left({n}−\mathrm{2}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)!} \\ $$$$=\mathrm{0} \\ $$$$ \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}\:=\:\mathrm{0} \\ $$$$ \\ $$$$\omega\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{0}+\mathrm{0}\right) \\ $$$$\therefore\:\omega\left(\mathrm{1}\right)=\mathrm{0} \\ $$

Commented by Filup last updated on 04/Dec/15

Can someone please show me why lim_(n→∞) (n/((n−1)!)) = 0

$$\mathrm{Can}\:\mathrm{someone}\:\mathrm{please}\:\mathrm{show}\:\mathrm{me}\:\mathrm{why} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{n}}{\left({n}−\mathrm{1}\right)!}\:=\:\mathrm{0} \\ $$

Commented by 123456 last updated on 04/Dec/15

lim_(n→∞) (n/((n−1)!))=lim_(n→∞) ((n−1+1)/((n−1)!)) =lim_(n→∞) (1/((n−2)!))+(1/((n−1)!)) =0

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}}{\left({n}−\mathrm{1}\right)!}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{{n}−\mathrm{1}+\mathrm{1}}{\left({n}−\mathrm{1}\right)!} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\left({n}−\mathrm{2}\right)!}+\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!} \\ $$$$=\mathrm{0} \\ $$

Commented by Filup last updated on 04/Dec/15

Nice Proof!

$$\mathcal{N}{ice}\:\mathcal{P}{roof}! \\ $$

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