Question

• Refer to the attachment.
• Simplify the solution.

Answer it correctly :)
No, irrelevent answer needed.
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Answer 1

$\sf\int\limits_1^{N+1}f(x)dx\leq \sum\limits_{k=1}^{N}f(k)\leq \int\limits_0^N f(x)dx$

$\sf\int\limits_1^{N+1}f(x)dx\leq \sum\limits_{k=1}^{N}f(k)\leq \int\limits_0^N f(x)dx$

• After substitutions N=n2, f(x)=n/(n2+x2) and simple computations we have

$\sf\arctan\frac{n^2+1}{n}-\arctan \frac{1}{n}\leq\sum\limits_{k=1}^{n^2}\sf\frac{n}{n^2+k^2}\leq\arctan narctan$

$\sf\arctan\frac{n^2+1}{n}-\arctan \frac{1}{n}\leq\sum\limits_{k=1}^{n^2}\frac{n}{n^2+k^2}\leq\arctan narctan$

Lets take a limit n→∞, then from sandwich lemma it follows

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

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