Question Number 159598 by Ar Brandon last updated on 19/Nov/21
$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{limits}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{sums}; \\ $$$${u}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{n}}{{nk}^{\mathrm{2}} +{k}+\mathrm{1}} \\ $$$${v}_{{n}} =\underset{\mathrm{1}\leqslant{k}\leqslant\mathrm{2}{n}} {\sum}\frac{{n}^{\mathrm{2}} }{{kn}^{\mathrm{2}} +{k}^{\mathrm{2}} } \\ $$$${w}_{{n}} =\underset{\mathrm{1}\leqslant{k}\leqslant{n}^{\mathrm{2}} } {\sum}\frac{\mathrm{sin}{k}}{{k}^{\mathrm{2}} }\left(\frac{{k}}{{k}+\mathrm{1}}\right)^{{n}} \\ $$