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A-n-n-n-2-1-2-n-n-2-2-2-n-n-2-n-2-Prove-lim-n-1-n-4-1-24-n-n-pi-4-A-n-1-4-2016-




Question Number 163608 by qaz last updated on 08/Jan/22
A_n =(n/(n^2 +1^2 ))+(n/(n^2 +2^2 ))+...+(n/(n^2 +n^2 ))  Prove:: lim_(n→∞) (1/(n^4 {(1/(24))−n[n((π/4)−A_n )−(1/4)]}))=2016
$$\mathrm{A}_{\mathrm{n}} =\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} }+\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} }+…+\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} +\mathrm{n}^{\mathrm{2}} } \\ $$$$\mathrm{Prove}::\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{4}} \left\{\frac{\mathrm{1}}{\mathrm{24}}−\mathrm{n}\left[\mathrm{n}\left(\frac{\pi}{\mathrm{4}}−\mathrm{A}_{\mathrm{n}} \right)−\frac{\mathrm{1}}{\mathrm{4}}\right]\right\}}=\mathrm{2016} \\ $$

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