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8. Sequences and Series
medium
If $y = x - {x^2} + {x^3} - {x^4} + ......\infty $, then value of $x$ will be
A
$y + \frac{1}{y}$
B
$\frac{y}{{1 + y}}$
C
$y - \frac{1}{y}$
D
$\frac{y}{{1 - y}}$
Solution
(d) $y = x – {x^2} + {x^3} – {x^4} + ……..\infty $
then $xy = {x^2} – {x^3} + {x^4} – ……\infty $
Adding, $y + xy = x + 0 + 0…… + 0$
$ \Rightarrow $$x – xy = y $
$\Rightarrow x(1 – y) = y$
$\Rightarrow x = \frac{y}{{1 – y}}$.
Aliter : $y = \frac{x}{{1 – ( – x)}} $
$\Rightarrow y = \frac{x}{{1 + x}}$
$ \Rightarrow $$y + yx = x$
$\Rightarrow x = \frac{y}{{1 – y}}$.
Standard 11
Mathematics