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Need to calculate the value of the following limit $$\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}$$ I don't know how. In general would know how to make that kind of limits with summations, where can I find examples and learn the topic, I appreciate any hint to make this limit.

Martin Sleziak
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Jhon Jairo
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1 Answers1

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By the Riemann sum

$$\lim_{n\rightarrow\infty}\sum_{k=0}^{n}\frac{k}{k^{2}+n^{2}}=\lim_{n\rightarrow\infty}\frac1n\sum_{k=0}^{n}\frac{\frac kn}{\left(\frac kn\right)^{2}+1}=\int_0^1\frac{x}{x^2+1}dx\\=\frac12\ln(x^2+1)\Bigg|_0^1=\frac{\ln2}{2}$$