If $Z_r=\cos \left(\frac{\pi}{2^r}\right)+i \sin \left(\frac{\pi}{2^r}\right)$ for $r=1,2,3, \ldots$. , then $Z_1 Z_2 Z_3 \ldots \infty$ equals to

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Aicharya · Accepted Answer

$\begin{aligned} & 	ext { Given, } z_r=\cos \left(\frac{\pi}{2^{\prime}}ight)+i \sin \left(\frac{\pi}{2^r}ight) \ & 	herefore z_1 z_2 z_3 \ldots=e^{i \frac{\pi}{2}} \cdot e^{i \frac{\pi}{2^2}} \cdot e^{i \frac{\pi}{2^3}} \ldots \ & =e^{i\left(\frac{\pi}{2}+\frac{\pi}{2^2}+\frac{\pi}{2^3}+\ldotsight)} \ & =e^{i\left(\frac{\pi / 2}{1}ight)}=e^{\lambda(\pi)} \ & =\cos \pi+i \sin \pi=-1+i(0) \ & =-1 \ & \end{aligned}$

If $Z_r=\cos \left(\frac{\pi}{2^r}\right)+i \sin \left(\frac{\pi}{2^r}\right)$ for $r=1,2,3, \ldots$. , then…

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