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

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

1. A
$y+\frac{1}{y}$
2. B
$\frac{y}{1-y}$
3. C
$y-\frac{1}{y}$
4. D
$\frac{y}{1+y}$

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### Solution:

$y=\alpha -{\alpha }^{2}+{\alpha }^{3}-{\alpha }^{4}+...\infty =\left(\alpha +{\alpha }^{3}+{\alpha }^{5}+...\infty \right)-\left({\alpha }^{2}+{\alpha }^{4}+{\alpha }^{6}+...\infty \right)$
$⇒y=\frac{\alpha }{1-{\alpha }^{2}}-\frac{{\alpha }^{2}}{1-{\alpha }^{2}}=\frac{\alpha -{\alpha }^{2}}{1-{\alpha }^{2}}=\frac{\alpha }{1+\alpha }$
$⇒\alpha =\frac{y}{1-y}$
Hnec, 2 is the correct option.

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