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Question Number 159598 by Ar Brandon last updated on 19/Nov/21

$$\mathrm{Determine}\mathrm{the}\mathrm{limits}\mathrm{of}\mathrm{the}\mathrm{following}\mathrm{sums};$$$$u_n=\underset{k=1}{\sum ^n}\frac{n}{{nk}^2+k+1}$$$$v_n=\sum _{1⩽k⩽2n}\frac{n^2}{{kn}^2+k^2}$$$$w_n=\sum _{1⩽k⩽n^2}\frac{\sin k}{k^2}{\left(\frac{k}{k+1}\right)}^n$$

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