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Find the value of α if |α| < 1 and $y = α - α^{2} + α^{3} - α^{4} + . . . \infty$ .

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y + \frac{1}{y}

\frac{y}{1 - y}

y - \frac{1}{y}

\frac{y}{1 + y}

answer is B.

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Detailed Solution

y = α - α^{2} + α^{3} - α^{4} + . . . \infty = (α + α^{3} + α^{5} + . . . \infty) - (α^{2} + α^{4} + α^{6} + . . . \infty)

\Rightarrow y = \frac{α}{1 - α^{2}} - \frac{α^{2}}{1 - α^{2}} = \frac{α - α^{2}}{1 - α^{2}} = \frac{α}{1 + α}

\Rightarrow α = \frac{y}{1 - y}

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