Find the sum of the series $4 - 5 x + 7 x^{2} - 9 x^{3} + 11 x^{4} - 13 x^{5} + . . . . . \infty$

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Solution

The correct option is A

Given series
s=4-5x+7 $x^{2} - 9 x^{3} + 11 x^{4} - 13 x^{5}$ +..... $\infty$ ------(1)
Multiply common ratio (-x) in equation 1

$- x s = - 4 x + 5 x^{2} - 7 x^{3} + 9 x^{4} - 11 x^{5} + . . . . . \infty$ ------(2)
Subtracting equation 2 from equation 1

$(1 + x) s = 4 - x + 2 x^{2} - 2 x^{3} + 2 x^{4} - 2 x^{5} + . . . . .$
$= 4 - x + 2 x^{2} (1 - x + x^{2} - . . . . \infty$

$(1 + x) s = 4 - x + \frac{2 x^{2}}{(x + 1)^{2}} = \frac{x^{2} + 3 x + 4}{1 + x}$

s= $\frac{x^{2} + 3 x + 4}{(x + 1)^{2}}$

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