If {x_n} = cos left( frac{pi }{3^n} right) + | vedclass

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(C) Given ${x_n} = \cos \left( \frac{\pi }{3^n} \right) + i\sin \left( \frac{\pi }{3^n} \right) = e^{i\pi / 3^n}$.The product is ${x_1} \cdot {x_2} \c

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$i$

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(C) Given ${x_n} = \cos \left( \frac{\pi }{3^n} \right) + i\sin \left( \frac{\pi }{3^n} \right) = e^{i\pi / 3^n}$.The product is ${x_1} \cdot {x_2} \cdot {x_3} \cdots {x_\infty } = e^{i\pi / 3^1} \cdot e^{i\pi / 3^2} \cdot e^{i\pi / 3^3} \cdots e^{i\pi / 3^\infty }$.Using the property of exponents,this equals $e^{i\pi \left( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \right)}$.The sum of the infinite geometric series in the exponent is $S = \frac{a}{1-r} = \frac{1/3}{1 - 1/3} = \frac{1/3}{2/3} = \frac{1}{2}$.Therefore,the product is $e^{i\pi (1/2)} = e^{i\pi / 2}$.Using Euler's formula,$e^{i\pi / 2} = \cos \left( \frac{\pi }{2} \right) + i\sin \left( \frac{\pi }{2} \right) = 0 + i(1) = i$.

If ${x_n} = \cos \left( \frac{\pi }{3^n} \right) + i\sin \left( \frac{\pi }{3^n} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots {x_\infty }$ is equal to:

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