left(cos frac{pi}{2}+i sin frac{pi}{2}right) | vedclass

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(C) Using the property of complex numbers in polar form,$e^{i	heta} = \cos 	heta + i \sin 	heta$,the given expression can be written as: $e^{i \fra

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$-1$

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(C) Using the property of complex numbers in polar form,$e^{i	heta} = \cos 	heta + i \sin 	heta$,the given expression can be written as: $e^{i \frac{\pi}{2}} \cdot e^{i \frac{\pi}{4}} \cdot e^{i \frac{\pi}{8}} \ldots \infty$ $= e^{i(\frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + \ldots \infty)}$ The exponent is an infinite geometric series with first term $a = \frac{\pi}{2}$ and common ratio $r = \frac{1}{2}$. The sum of an infinite geometric series is $S_{\infty} = \frac{a}{1-r}$. $S_{\infty} = \frac{\frac{\pi}{2}}{1 - \frac{1}{2}} = \frac{\frac{\pi}{2}}{\frac{1}{2}} = \pi$. Thus,the expression becomes $e^{i\pi}$. Using Euler's formula,$e^{i\pi} = \cos \pi + i \sin \pi = -1 + i(0) = -1$. Hence,option $C$ is correct.

$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \times \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \infty =$

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