How to show convergence of : $$\lim_{n \to \infty}\sum_{k=1}^{n} \frac{n+k}{(n^2+k^2)}$$
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Hi. Tried helping with the typesetting, please let me know if I got it wrong. – mathreadler Apr 23 '17 at 09:31
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Use the idea of https://math.stackexchange.com/questions/469885/the-limit-of-a-sum-sum-k-1n-fracnn2k2/469886#469886 – lab bhattacharjee Apr 23 '17 at 09:32
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1Maybe there should be a limit $n \to \infty$? – mathreadler Apr 23 '17 at 09:37
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Thank you,I'll try solving it that way,the problem asks me to prove that this series converges – Lola Apr 23 '17 at 09:38
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1Do you mean the series with general term this sum, or the sequence defined by this sum? – Bernard Apr 23 '17 at 09:58
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Indeed, which series? – Did Apr 23 '17 at 10:07
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We will try to evaluate it using limit of a sum which will eventually prove that it converges.
$$\lim_{n \to \infty} \sum_{k=1}^n \frac{(n+k)\frac{1}{n^2}}{(n^2+k^2)\frac{1}{n^2}}$$ $$\lim_{n \to \infty} \sum_{k=1}^n \frac{(1+\frac{k}{n})\frac{1}{n}}{(1+\frac{k^2}{n^2})}$$ Converting it into a definite integral. $$\int_{0}^1 \frac{1+x}{1+x^2} dx$$ $$\int_{0}^1 \left(\frac{1}{1+x^2}+\frac{x}{1+x^2}\right) dx$$ This can be easily evaluated.
The result comes out to be $\frac{\pi}{4} + \frac{1}{2}ln2$
Which proves that it the given sum converges.
Aman Sharma
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