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Question Number 30757 by abdo imad last updated on 25/Feb/18

$$find{lim}_{n\to \mathrm{\infty }}(\frac{1}{n}+\frac{1}{\sqrt{n^2-1}}+....+\frac{1}{\sqrt{n^2-{(n-1)}^2}})$$

Commented by abdo imad last updated on 28/Feb/18

$$letput{w}_n=\frac{1}{n}+\frac{1}{\sqrt{n^2-1}}+...+\frac{1}{\sqrt{n^2-{(n-1)}^2}}$$$$w_n=\frac{1}{n}(1+\frac{1}{\sqrt{1-\frac{1^2}{n^2}}}+\frac{1}{\sqrt{1-\frac{2^2}{n^2}}}+.....+\frac{1}{\sqrt{1-\frac{{(n-1)}^2}{n^2}}})$$$$=\frac{1}{n}\sum _{k=0}^{n-1}\frac{1}{\sqrt{1-{\left(\frac{k}{n}\right)}^2}}so{w}_{n}isaRiemansumand$$$${lim}_{n\to \mathrm{\infty }}{w}_n={\int }_0^1\frac{dx}{\sqrt{1-x^2}}={\left[arcsinx\right]}_0^1=\frac{\pi }{2}.$$

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