MathematicsFind the value of α if |α| < 1 and y=α-α2+α3-α4+...∞.

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

A
$y + \frac{1}{y}$
B
$\frac{y}{1 - y}$
C
$y - \frac{1}{y}$
D
$\frac{y}{1 + y}$

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}

Hnec, 2 is the correct option.

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