Let alpha, beta denote the cube roots of unit | vedclass

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(A) The cube roots of unity other than $1$ are $\omega$ and $\omega^{2}$, where $\omega = e^{i 2\pi/3}$.Given $S = \sum_{n=0}^{\infty} (-1)^{n} \left(

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either $-2 \omega$ or $-2 \omega^{2}$

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(A) The cube roots of unity other than $1$ are $\omega$ and $\omega^{2}$, where $\omega = e^{i 2\pi/3}$.Given $S = \sum_{n=0}^{\infty} (-1)^{n} \left(\frac{\alpha}{\beta}\right)^{n}$, this is a geometric series with first term $a = 1$ and common ratio $r = -\frac{\alpha}{\beta}$.The sum of an infinite geometric series is given by $S = \frac{a}{1-r} = \frac{1}{1 - (-\alpha/\beta)} = \frac{\beta}{\alpha + \beta}$.Since $1 + \omega + \omega^{2} = 0$, we have $\alpha + \beta = \omega + \omega^{2} = -1$.Case $I$: If $\alpha = \omega$ and $\beta = \omega^{2}$, then $S = \frac{\omega^{2}}{\omega + \omega^{2}} = \frac{\omega^{2}}{-1} = -\omega^{2}$.Case $II$: If $\alpha = \omega^{2}$ and $\beta = \omega$, then $S = \frac{\omega}{\omega^{2} + \omega} = \frac{\omega}{-1} = -\omega$.Thus, the value of $S$ is either $-\omega$ or $-\omega^{2}$.Note: The provided options seem to contain a factor of $2$ which is not present in the standard sum of this infinite series. Based on the structure, the intended answer is $-\omega$ or $-\omega^{2}$.

Let $\alpha, \beta$ denote the cube roots of unity other than $1$ and $\alpha \neq \beta$. Let $S = \sum_{n=0}^{\infty} (-1)^{n} \left(\frac{\alpha}{\beta}\right)^{n}$. Then the value of $S$ is

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