If $x_n=\cos \left(\frac{\pi}{4^n}\right)+i \sin \left(\frac{\pi}{4^n}\right)$, then $x_1 x_2 x_3 \ldots \infty$ is equal to

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Aicharya · Accepted Answer

We have, $ \begin{aligned} x_n & =e^i \frac{\pi}{4 n} \ x_1 x_2 x_3 \ldots & =e^i \frac{\pi}{4} \cdot e^i \frac{\pi}{4^2} \cdot e^i \frac{\pi}{4^3} \ldots \ & =e^{i \pi}\left(\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3} \ldotsight) \ =e^{i \pi} \frac{1 / 4}{1-\frac{1}{4}} & =e^i \frac{\pi}{3}=\cos 60^{\circ}+i \sin 60^{\circ} \ =\frac{1}{2}+\frac{i \sqrt{3}}{2} & =\frac{1+i \sqrt{3}}{2} \end{aligned} $

If $x_n=\cos \left(\frac{\pi}{4^n}\right)+i \sin \left(\frac{\pi}{4^n}\right)$, then $x_1 x_2 x_3 \ldots…

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