Problem
Evaluate $$\lim_{n \to \infty}\left[\sqrt{\frac{1\cdot2}{n^2+1}}+\sqrt{\frac{2\cdot3}{n^2+2}}+\cdots+\sqrt{\frac{n(n+1)}{n^2+n}}\right],n=1,2,\cdots.$$
Solution
Notice that, for $k=1,2,\cdots,n$, $$ \sqrt{\frac{k\cdot(k+1)}{n^2+k}} \geq \frac{k}{\sqrt{n^2+n}}\geq \frac{k}{\sqrt{n^2+n^2}}=\frac{k}{n\sqrt{2}}$$ Therefore $$\sum_{k=1}^{n}\sqrt{\frac{k\cdot(k+1)}{n^2+k}}\geq \sum_{k=1}^{n}\frac{k}{n\sqrt{2}}=\frac{n+1}{2\sqrt{2}}\to+\infty(n \to \infty).$$ As a result $$\lim_{n \to \infty}\sum_{k=1}^{n}\sqrt{\frac{k\cdot(k+1)}{n^2+k}}=+\infty.$$
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