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For-k-lt-N-fixed-and-gt-0-then-lim-n-1-n-i-1-k-n-k-i-i-1-k-n-i-n-




Question Number 150548 by mathdanisur last updated on 13/Aug/21
For  k<N  fixed  and  š›‚>0  then:  lim_(nā†’āˆž) (1/( (āˆšn^š›‚ ))) āˆ™ (((Ī _(i=1) ^k (n+k+i))/(Ī _(i=1) ^k (n+i))))^n^š›‚
$$\mathrm{For}\:\:\boldsymbol{\mathrm{k}}<\mathbb{N}\:\:\mathrm{fixed}\:\:\mathrm{and}\:\:\boldsymbol{\alpha}>\mathrm{0}\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}^{\boldsymbol{\alpha}} }}\:\centerdot\:\left(\frac{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{k}+\mathrm{i}\right)}{\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\prod}}\left(\mathrm{n}+\mathrm{i}\right)}\right)^{\boldsymbol{\mathrm{n}}^{\boldsymbol{\alpha}} } \\ $$

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