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Question Number 66341 by mathmax by abdo last updated on 12/Aug/19

$$find{lim}_{n\to +\mathrm{\infty }}\frac{1}{n^4}\sum _{k=1}^n\frac{k^3}{\sqrt{{(1+{\left(\frac{k}{n}\right)}^2)}^3}}$$

Commented by Prithwish sen last updated on 13/Aug/19

$$\frac{1}{\mathrm{n}}{\lim }_{\mathrm{n}\to \mathrm{\infty }}\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{{\left(\frac{\mathrm{k}}{\mathrm{n}}\right)}^3}{\sqrt{{(1+{\left(\frac{\mathrm{k}}{\mathrm{n}}\right)}^2)}^3}}={\int }_0^1\frac{{\mathrm{x}}^3}{\sqrt{{(1+{\mathrm{x}}^2)}^3}}\mathrm{dx}$$$$={\left[\frac{2+{\mathrm{x}}^2}{\sqrt{1+{\mathrm{x}}^2}}\right]}_0^1=\frac{3-2\sqrt{2}}{\sqrt{2}}$$

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