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It's the first time that I'm in front of a limit of this kind: i mean with a summation inside. It must solve as a summation?. I have no idea how to solve it. Could you please help me? Thanks $$\lim _{n\to \infty }\left(\sum _{i=1}^n\:\left(\sqrt{1+\frac{2i}{n}}\right)\frac{2}{n}\right)$$

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$$2\int_{0}^{1}\sqrt{1+2x}\,dx = \int_{0}^{2}\sqrt{1+z}\,dz = \left.\frac{2}{3}(1+z)^{3/2}\right|_{0}^{2}=\color{red}{2\sqrt{3}-\frac{2}{3}.}$$ Your sum is a Riemann sum associated with the integral in the LHS.

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