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Question Number 128846 by Ar Brandon last updated on 10/Jan/21

$${\mathrm{u}}_{\mathrm{n}}=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\sin \left(\frac{\mathrm{k}\pi }{\mathrm{n}}\right)\sin \left(\frac{\mathrm{k}}{{\mathrm{n}}^2}\right)$$$$\text{Double subscripts: use braces to clarify}$$

Commented by Dwaipayan Shikari last updated on 10/Jan/21

$$\underset{n\to \mathrm{\infty }}{\lim }sin\left(\frac{k}{n^2}\right)=\frac{k}{n^2}$$$$u_n=\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\underset{k=1}{\sum ^n}\frac{k}{n}sin\frac{k\pi }{n}$$$$={\int }_0^{1}xsinx\pi dx=\frac{1}{{\pi }^2}{\int }_0^{\pi }usinu=\frac{1}{{\pi }^2}{[-ucosu]}_0^{\pi }+\frac{1}{{\pi }^2}{\int }_0^{\pi }cosudu$$$$=\frac{1}{\pi }$$

Commented by Dwaipayan Shikari last updated on 10/Jan/21

$$\frac{1}{\pi }=\frac{2\sqrt{2}}{9801}\underset{k=0}{\sum ^{\mathrm{\infty }}}\frac{\left(4k\right)!(1103+26390k)}{{(k!)}^4{396}^{4k}}$$

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