Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 181183 by Shrinava last updated on 22/Nov/22

Ω_n = determinant ((1,1,1,(...),1),(1,2^2 ,2^3 ,(...),2^n ),(1,3^2 ,3^3 ,(...),3^n ),((...),(...),(...),(...),(...)),(1,n^2 ,n^3 ,(...),n^n )) , n ∈ N^∗ Find: Ω =lim_(n→∞) ((Ω_(n+1) /Ω_n ))^(1/n)

$$\Omega_{\boldsymbol{\mathrm{n}}} \:=\:\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}&{...}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{2}^{\mathrm{2}} }&{\mathrm{2}^{\mathrm{3}} }&{...}&{\mathrm{2}^{\boldsymbol{\mathrm{n}}} }\\{\mathrm{1}}&{\mathrm{3}^{\mathrm{2}} }&{\mathrm{3}^{\mathrm{3}} }&{...}&{\mathrm{3}^{\boldsymbol{\mathrm{n}}} }\\{...}&{...}&{...}&{...}&{...}\\{\mathrm{1}}&{\mathrm{n}^{\mathrm{2}} }&{\mathrm{n}^{\mathrm{3}} }&{...}&{\mathrm{n}^{\boldsymbol{\mathrm{n}}} }\end{vmatrix}\:\:,\:\:\:\mathrm{n}\:\in\:\mathbb{N}^{\ast} \\ $$$$\mathrm{Find}:\:\:\:\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{\boldsymbol{\mathrm{n}}}]{\frac{\Omega_{\boldsymbol{\mathrm{n}}+\mathrm{1}} }{\Omega_{\boldsymbol{\mathrm{n}}} }}\: \\ $$

Answered by aleks041103 last updated on 22/Nov/22

the determiant D(x_i ): D(x_i )= determinant ((1,1,(...),1),(x_1 ^2 ,x_2 ^2 ,(...),x_n ^2 ),(x_1 ^3 ,x_2 ^3 ,(...),x_n ^3 ),((...),(...),(...),(...)),(x_1 ^n ,x_2 ^n ,(...),x_n ^n )) by laplace formula D=ε_(i_1 i_2 ...i_n ) x_i_1 ^0 x_i_2 ^2 x_i_3 ^3 ...x_i_n ^n = =ε_(i_1 ...i_n ) x_i_2 ^2 ...x_i_n ^2 (x_i_2 ^0 x_i_3 ^1 ...x_i_n ^(n−2) )= =(x_1 x_2 ...x_n )^2 ε_(i_1 ...i_n ) (1/x_i_1 ^2 )(x_i_2 ^0 x_i_3 ^1 ...x_i_n ^(n−2) ) ε_(k σ(12...(k−1)(k+1)...n)) =(−1)^(sgn(σ)) ε_(k12...(k−1)(k+1)...n) = =(−1)^(sgn(σ)) (−1)^(k−1) ε_(12...(k−1)k(k+1)...n) = =(−1)^(k−1) ε_(σ(12..(k−1)(k+1)..n)) ⇒ε_(i_1 ...i_n ) =(−1)^(i_1 −1) ε_(i_2 ...i_n ) ⇒D=(x_1 x_2 ...x_n )^2 Σ_(s=1) ^n (((−1)^(s−1) )/x_s ^2 )(ε_(i_2 ...i_n ) x_i_2 ^0 x_i_3 ^1 ...x_i_n ^(n−2) )_(i_j ≠s) (ε_(i_2 ...i_n ) x_i_2 ^0 x_i_3 ^1 ...x_i_n ^(n−2) )_(i_j ≠s) = determinant ((1,1,(...),1,1,(...),1),(x_1 ,x_2 ,(...),x_(s−1) ,x_(s+1) ,(...),x_n ),((...),(...),(...),(...),(...),(...),(...)),(x_1 ^(n−2) ,x_2 ^(n−2) ,(...),x_(s−1) ^(n−2) ,x_(s+1) ^(n−2) ,(...),x_n ^(n−2) ))= =Π_(i>j(i,j≠s)) (x_i −x_j ) ⇒D=(x_1 x_2 ...x_n )^2 Σ_(s=1) ^n (((−1)^(s−1) )/x_s ^2 )Π_(i>j(i,j≠s)) (x_i −x_j ) In our case: x_i =i ⇒Ω_n =(n!)^2 Σ_(s=1) ^n (((−1)^(s−1) )/s^2 )Π_(i>j(i,j≠s)) (i−j) Π_(i>j(i,j≠s)) (i−j)=((Π_(i>j) (i−j))/((Π_(s>j) (s−j))(Π_(i>s) (i−s)))) Π_(s>j) (s−j)=Π_(j=1) ^(s−1) (s−j)=(s−1)(s−2)...2.1=(s−1)! Π_(i>s) (i−s)=Π_(i=s+1) ^n (i−s)=1.2...(n−s)=(n−s)! Π_(i>j) (i−j)=Π_(j=1) ^(n−1) Π_(i=j+1) ^n (i−j)= =Π_(j=1) ^(n−1) Π_(i=1) ^(n−j) i=Π_(j=1) ^(n−1) (n−j)!=Π_(j=1) ^(n−1) j! ⇒Ω_n =(Π_(j=1) ^(n−1) j!)(n!)^2 Σ_(s=1) ^n (((−1)^(s−1) )/s^2 ) (1/((s−1)!(n−s)!))= =(Π_(j=1) ^n j!)Σ_(s=1) ^n ((n),(s) ) (((−1)^(s−1) )/s) (1−x)^n =Σ_(s=0) ^n ((n),(s) ) (−1)^s x^s =1−xΣ_(s=1) ^n ((n),(s) ) (−1)^(s−1) x^(s−1) ⇒Σ_(s=1) ^n ((n),(s) ) (−1)^(s−1) x^(s−1) =((1−(1−x)^n )/x) ⇒∫_0 ^1 (Σ_(s=1) ^n ((n),(s) ) (−1)^(s−1) x^(s−1) )dx= =Σ_(s=1) ^n ((n),(s) ) (−1)^(s−1) (∫_0 ^1 x^(s−1) dx)= =Σ_(s=1) ^n ((n),(s) ) (((−1)^s )/s)=∫_0 ^1 ((1−(1−x)^n )/x)dx ∫_0 ^1 ((1−(1−x)^n )/x)dx=∫_0 ^1 ((1−x^n )/(1−x))dx=∫_0 ^1 Σ_(k=0) ^(n−1) x^k dx= =Σ_(k=0) ^(n−1) ∫_0 ^1 x^k dx=Σ_(k=0) ^(n−1) (1/(k+1))=Σ_(k=1) ^n (1/k)=H(n) ⇒Ω_n =(1!2!...n!)H(n) (Ω_(n+1) /Ω_n )=a_n =(((1!2!...n!(n+1)!)H(n+1))/((1!2!...n!)H(n)))=(n+1)!((H(n+1))/(H(n))) L=lim_(n→∞) (a_n )^(1/n) =lim_(n→∞) (a_(n+1) /a_n )= =lim_(n→∞) (((n+2)!H(n+2)H(n))/((n+1)!(H(n+1))^2 ))=lim_(n→∞) (n+2)((H(n)H(n+2))/((H(n+1))^2 )) H(n)∼ln(n) ⇒L=lim_(n→∞) (((n+2)ln(n)ln(n+2))/(ln(n+1)ln(n+1))) but lim_(n→∞) ((ln(n))/(ln(n+1)))=1⇒ L=lim_(n→∞) (n+2)=∞ ⇒lim_(n→∞) ((Ω_(n+1) /Ω_n ))^(1/n) →∞

$${the}\:{determiant}\:{D}\left({x}_{{i}} \right): \\ $$$${D}\left({x}_{{i}} \right)=\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{...}&{\mathrm{1}}\\{{x}_{\mathrm{1}} ^{\mathrm{2}} }&{{x}_{\mathrm{2}} ^{\mathrm{2}} }&{...}&{{x}_{{n}} ^{\mathrm{2}} }\\{{x}_{\mathrm{1}} ^{\mathrm{3}} }&{{x}_{\mathrm{2}} ^{\mathrm{3}} }&{...}&{{x}_{{n}} ^{\mathrm{3}} }\\{...}&{...}&{...}&{...}\\{{x}_{\mathrm{1}} ^{{n}} }&{{x}_{\mathrm{2}} ^{{n}} }&{...}&{{x}_{{n}} ^{{n}} }\end{vmatrix} \\ $$$${by}\:{laplace}\:{formula} \\ $$$${D}=\epsilon_{{i}_{\mathrm{1}} {i}_{\mathrm{2}} ...{i}_{{n}} } {x}_{{i}_{\mathrm{1}} } ^{\mathrm{0}} {x}_{{i}_{\mathrm{2}} } ^{\mathrm{2}} {x}_{{i}_{\mathrm{3}} } ^{\mathrm{3}} ...{x}_{{i}_{{n}} } ^{{n}} = \\ $$$$=\epsilon_{{i}_{\mathrm{1}} ...{i}_{{n}} } {x}_{{i}_{\mathrm{2}} } ^{\mathrm{2}} ...{x}_{{i}_{{n}} } ^{\mathrm{2}} \left({x}_{{i}_{\mathrm{2}} } ^{\mathrm{0}} {x}_{{i}_{\mathrm{3}} } ^{\mathrm{1}} ...{x}_{{i}_{{n}} } ^{{n}−\mathrm{2}} \right)= \\ $$$$=\left({x}_{\mathrm{1}} {x}_{\mathrm{2}} ...{x}_{{n}} \right)^{\mathrm{2}} \epsilon_{{i}_{\mathrm{1}} ...{i}_{{n}} } \frac{\mathrm{1}}{{x}_{{i}_{\mathrm{1}} } ^{\mathrm{2}} }\left({x}_{{i}_{\mathrm{2}} } ^{\mathrm{0}} {x}_{{i}_{\mathrm{3}} } ^{\mathrm{1}} ...{x}_{{i}_{{n}} } ^{{n}−\mathrm{2}} \right) \\ $$$$\epsilon_{{k}\:\sigma\left(\mathrm{12}...\left({k}−\mathrm{1}\right)\left({k}+\mathrm{1}\right)...{n}\right)} =\left(−\mathrm{1}\right)^{{sgn}\left(\sigma\right)} \epsilon_{{k}\mathrm{12}...\left({k}−\mathrm{1}\right)\left({k}+\mathrm{1}\right)...{n}} = \\ $$$$=\left(−\mathrm{1}\right)^{{sgn}\left(\sigma\right)} \left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \epsilon_{\mathrm{12}...\left({k}−\mathrm{1}\right){k}\left({k}+\mathrm{1}\right)...{n}} = \\ $$$$=\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \epsilon_{\sigma\left(\mathrm{12}..\left({k}−\mathrm{1}\right)\left({k}+\mathrm{1}\right)..{n}\right)} \\ $$$$\Rightarrow\epsilon_{{i}_{\mathrm{1}} ...{i}_{{n}} } =\left(−\mathrm{1}\right)^{{i}_{\mathrm{1}} −\mathrm{1}} \epsilon_{{i}_{\mathrm{2}} ...{i}_{{n}} } \\ $$$$\Rightarrow{D}=\left({x}_{\mathrm{1}} {x}_{\mathrm{2}} ...{x}_{{n}} \right)^{\mathrm{2}} \underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} }{{x}_{{s}} ^{\mathrm{2}} }\left(\epsilon_{{i}_{\mathrm{2}} ...{i}_{{n}} } {x}_{{i}_{\mathrm{2}} } ^{\mathrm{0}} {x}_{{i}_{\mathrm{3}} } ^{\mathrm{1}} ...{x}_{{i}_{{n}} } ^{{n}−\mathrm{2}} \right)_{{i}_{{j}} \neq{s}} \\ $$$$\left(\epsilon_{{i}_{\mathrm{2}} ...{i}_{{n}} } {x}_{{i}_{\mathrm{2}} } ^{\mathrm{0}} {x}_{{i}_{\mathrm{3}} } ^{\mathrm{1}} ...{x}_{{i}_{{n}} } ^{{n}−\mathrm{2}} \right)_{{i}_{{j}} \neq{s}} =\begin{vmatrix}{\mathrm{1}}&{\mathrm{1}}&{...}&{\mathrm{1}}&{\mathrm{1}}&{...}&{\mathrm{1}}\\{{x}_{\mathrm{1}} }&{{x}_{\mathrm{2}} }&{...}&{{x}_{{s}−\mathrm{1}} }&{{x}_{{s}+\mathrm{1}} }&{...}&{{x}_{{n}} }\\{...}&{...}&{...}&{...}&{...}&{...}&{...}\\{{x}_{\mathrm{1}} ^{{n}−\mathrm{2}} }&{{x}_{\mathrm{2}} ^{{n}−\mathrm{2}} }&{...}&{{x}_{{s}−\mathrm{1}} ^{{n}−\mathrm{2}} }&{{x}_{{s}+\mathrm{1}} ^{{n}−\mathrm{2}} }&{...}&{{x}_{{n}} ^{{n}−\mathrm{2}} }\end{vmatrix}= \\ $$$$=\underset{{i}>{j}\left({i},{j}\neq{s}\right)} {\prod}\left({x}_{{i}} −{x}_{{j}} \right) \\ $$$$\Rightarrow{D}=\left({x}_{\mathrm{1}} {x}_{\mathrm{2}} ...{x}_{{n}} \right)^{\mathrm{2}} \underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} }{{x}_{{s}} ^{\mathrm{2}} }\underset{{i}>{j}\left({i},{j}\neq{s}\right)} {\prod}\left({x}_{{i}} −{x}_{{j}} \right) \\ $$$${In}\:{our}\:{case}:\:{x}_{{i}} ={i} \\ $$$$\Rightarrow\Omega_{{n}} =\left({n}!\right)^{\mathrm{2}} \underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} }{{s}^{\mathrm{2}} }\underset{{i}>{j}\left({i},{j}\neq{s}\right)} {\prod}\left({i}−{j}\right) \\ $$$$\underset{{i}>{j}\left({i},{j}\neq{s}\right)} {\prod}\left({i}−{j}\right)=\frac{\underset{{i}>{j}} {\prod}\left({i}−{j}\right)}{\left(\underset{{s}>{j}} {\prod}\left({s}−{j}\right)\right)\left(\underset{{i}>{s}} {\prod}\left({i}−{s}\right)\right)} \\ $$$$\underset{{s}>{j}} {\prod}\left({s}−{j}\right)=\underset{{j}=\mathrm{1}} {\overset{{s}−\mathrm{1}} {\prod}}\left({s}−{j}\right)=\left({s}−\mathrm{1}\right)\left({s}−\mathrm{2}\right)...\mathrm{2}.\mathrm{1}=\left({s}−\mathrm{1}\right)! \\ $$$$\underset{{i}>{s}} {\prod}\left({i}−{s}\right)=\underset{{i}={s}+\mathrm{1}} {\overset{{n}} {\prod}}\left({i}−{s}\right)=\mathrm{1}.\mathrm{2}...\left({n}−{s}\right)=\left({n}−{s}\right)! \\ $$$$\underset{{i}>{j}} {\prod}\left({i}−{j}\right)=\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\underset{{i}={j}+\mathrm{1}} {\overset{{n}} {\prod}}\left({i}−{j}\right)= \\ $$$$=\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\underset{{i}=\mathrm{1}} {\overset{{n}−{j}} {\prod}}{i}=\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}\left({n}−{j}\right)!=\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{j}! \\ $$$$\Rightarrow\Omega_{{n}} =\left(\underset{{j}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{j}!\right)\left({n}!\right)^{\mathrm{2}} \underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} }{{s}^{\mathrm{2}} }\:\frac{\mathrm{1}}{\left({s}−\mathrm{1}\right)!\left({n}−{s}\right)!}= \\ $$$$=\left(\underset{{j}=\mathrm{1}} {\overset{{n}} {\prod}}{j}!\right)\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\frac{\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} }{{s}} \\ $$$$\left(\mathrm{1}−{x}\right)^{{n}} =\underset{{s}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{s}} {x}^{{s}} =\mathrm{1}−{x}\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} {x}^{{s}−\mathrm{1}} \\ $$$$\Rightarrow\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} {x}^{{s}−\mathrm{1}} =\frac{\mathrm{1}−\left(\mathrm{1}−{x}\right)^{{n}} }{{x}} \\ $$$$\Rightarrow\int_{\mathrm{0}} ^{\mathrm{1}} \left(\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} {x}^{{s}−\mathrm{1}} \right){dx}= \\ $$$$=\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\left(−\mathrm{1}\right)^{{s}−\mathrm{1}} \left(\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{s}−\mathrm{1}} {dx}\right)= \\ $$$$=\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{s}}\end{pmatrix}\:\frac{\left(−\mathrm{1}\right)^{{s}} }{{s}}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\left(\mathrm{1}−{x}\right)^{{n}} }{{x}}{dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−\left(\mathrm{1}−{x}\right)^{{n}} }{{x}}{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}{dx}=\int_{\mathrm{0}} ^{\mathrm{1}} \underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}{x}^{{k}} {dx}= \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{k}} {dx}=\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{\mathrm{1}}{{k}+\mathrm{1}}=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}={H}\left({n}\right) \\ $$$$\Rightarrow\Omega_{{n}} =\left(\mathrm{1}!\mathrm{2}!...{n}!\right){H}\left({n}\right) \\ $$$$\frac{\Omega_{{n}+\mathrm{1}} }{\Omega_{{n}} }={a}_{{n}} =\frac{\left(\mathrm{1}!\mathrm{2}!...{n}!\left({n}+\mathrm{1}\right)!\right){H}\left({n}+\mathrm{1}\right)}{\left(\mathrm{1}!\mathrm{2}!...{n}!\right){H}\left({n}\right)}=\left({n}+\mathrm{1}\right)!\frac{{H}\left({n}+\mathrm{1}\right)}{{H}\left({n}\right)} \\ $$$${L}=\underset{{n}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{{a}_{{n}} }=\underset{{n}\rightarrow\infty} {{lim}}\frac{{a}_{{n}+\mathrm{1}} }{{a}_{{n}} }= \\ $$$$=\underset{{n}\rightarrow\infty} {{lim}}\frac{\left({n}+\mathrm{2}\right)!{H}\left({n}+\mathrm{2}\right){H}\left({n}\right)}{\left({n}+\mathrm{1}\right)!\left({H}\left({n}+\mathrm{1}\right)\right)^{\mathrm{2}} }=\underset{{n}\rightarrow\infty} {{lim}}\left({n}+\mathrm{2}\right)\frac{{H}\left({n}\right){H}\left({n}+\mathrm{2}\right)}{\left({H}\left({n}+\mathrm{1}\right)\right)^{\mathrm{2}} } \\ $$$${H}\left({n}\right)\sim{ln}\left({n}\right) \\ $$$$\Rightarrow{L}=\underset{{n}\rightarrow\infty} {{lim}}\frac{\left({n}+\mathrm{2}\right){ln}\left({n}\right){ln}\left({n}+\mathrm{2}\right)}{{ln}\left({n}+\mathrm{1}\right){ln}\left({n}+\mathrm{1}\right)} \\ $$$${but}\:\underset{{n}\rightarrow\infty} {{lim}}\frac{{ln}\left({n}\right)}{{ln}\left({n}+\mathrm{1}\right)}=\mathrm{1}\Rightarrow \\ $$$${L}=\underset{{n}\rightarrow\infty} {{lim}}\left({n}+\mathrm{2}\right)=\infty \\ $$$$\Rightarrow\underset{{n}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\frac{\Omega_{{n}+\mathrm{1}} }{\Omega_{{n}} }}\rightarrow\infty \\ $$

Commented by Shrinava last updated on 23/Nov/22

Thank you so much my dear professor, perfect solution as always

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{professor}, \\ $$$$\mathrm{perfect}\:\mathrm{solution}\:\mathrm{as}\:\mathrm{always} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com