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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.

MathsRookie
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    Write $\frac{k}{n^2+k^2}=\frac{1}{n}\frac{k/n}{1+(k/n)^2}$ and see that your sum is a Riemann sum on $[0,1]$ for the uniform partition. –  Jul 03 '18 at 11:24
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    Also: https://math.stackexchange.com/q/2206368/42969, https://math.stackexchange.com/q/1446408/42969, https://math.stackexchange.com/q/692149/42969 – all found with Approach0 – Martin R Jul 03 '18 at 11:42

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

Nosrati
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