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Question Number 133181 by mnjuly1970 last updated on 19/Feb/21

$$......nicecalculus...$$$$lim{}_{n\to \mathrm{\infty }}\left\{n\underset{k=1}{\sum ^n}{\left(\frac{1}{n+k}\right)}^2\right\}=??$$

Answered by Dwaipayan Shikari last updated on 19/Feb/21

$$\underset{n\to \mathrm{\infty }}{\lim }n\underset{k=1}{\sum ^n}\frac{1}{{(n+k)}^2}=\frac{1}{n}\underset{k=1}{\sum ^n}\frac{1}{{(1+\frac{k}{n})}^2}={\int }_0^1\frac{1}{{(1+x)}^2}dx=\frac{1}{2}$$

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