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Question Number 99803 by student work last updated on 23/Jun/20

$$\underset{\mathrm{k}=1}{\prod ^{\mathrm{\infty }}}(1+\frac{1}{{\mathrm{k}}^2})=?$$$$\mathrm{helpe}\mathrm{me}$$

Answered by smridha last updated on 23/Jun/20

$$\mathit{w}\mathit{e}\mathit{k}\mathit{n}\mathit{o}\mathit{w}\mathit{s}\mathit{i}\mathit{n}\left(\mathit{z}\right)=\mathit{z}\underset{\mathit{k}=1}{\prod ^{\mathrm{\infty }}}[1-{\left(\frac{\mathit{z}}{\mathit{k}\mathit{\pi }}\right)}^2]$$$$\mathit{s}\mathit{o}\underset{\mathit{k}=1}{\prod ^{\mathrm{\infty }}}(1+\frac{1}{{\mathit{k}}^2})=\underset{\mathit{z}\to \mathit{i}\mathit{\pi }}{\lim }\frac{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{z}\right)}{\mathit{z}}$$$$=\underset{\mathit{z}\to \mathit{i}\mathit{\pi }}{\lim }\frac{{\mathit{e}}^{\mathit{i}\mathit{z}}-{\mathit{e}}^{-\mathit{i}\mathit{z}}}{2\mathit{i}\mathit{z}}$$$$=\frac{{\mathit{e}}^{-\mathit{\pi }}-{\mathit{e}}^{\mathit{\pi }}}{-2\mathit{\pi }}=\frac{1}{\mathit{\pi }}\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(\mathit{\pi }\right)\approx 3.676....$$

Answered by mathmax by abdo last updated on 23/Jun/20

$$\mathrm{we}\mathrm{have}\sin \left(\pi \mathrm{z}\right)=\pi \mathrm{z}\prod _{\mathrm{n}=1}^{\mathrm{\infty }}(1-\frac{{\mathrm{z}}^2}{{\mathrm{n}}^2})$$$$\mathrm{z}=\mathrm{i}\Rightarrow \sin \left(\mathrm{i}\pi \right)=\mathrm{i}\pi \prod _{\mathrm{n}=1}^{\mathrm{\infty }}(1+\frac{1}{{\mathrm{n}}^2})\Rightarrow \prod _{\mathrm{n}=1}^{\mathrm{\infty }}(1+\frac{1}{{\mathrm{n}}^2})=\frac{\sin \left(\mathrm{i}\pi \right)}{\mathrm{i}\pi }$$$$=\frac{{\mathrm{e}}^{-\pi }-{\mathrm{e}}^{\pi }}{2\mathrm{i}\times \left(\mathrm{i}\pi \right)}=\frac{{\mathrm{e}}^{\pi }-{\mathrm{e}}^{-\pi }}{2}\times \frac{1}{\pi }=\frac{\mathrm{sh}\left(\pi \right)}{\pi }$$$$$$

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