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I haven't touched series and sequences in a while and am rusty, so need some direction in finding the following:

$$ \lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}$$

Thanks in advance.

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marked as duplicate by Martin R, Community Jul 3 '18 at 12:01

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Use Riemann sum: $$\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{k}{n^2 + k^2}=\lim\limits_{n \to \infty} \sum_{k=1}^n \frac{\frac{k}{n}}{1 + \left(\frac{k}{n}\right)^2}\frac{1}{n}=\int_0^1\frac{x}{1+x^2}dx$$

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