Sum $1 + 3 x + 5 x^{2} + 7 x^{3} + 9 x^{4} + . . . . .$ to infinity is $(x < 1)$

\frac{1 + x}{{(1 - x)}^{2}}

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\frac{1 - x}{{(1 + x)}^{2}}

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\frac{1 - x}{(1 + x)}

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\frac{1 + x}{(1 - x)}

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Solution

The correct option is A $\frac{1 + x}{{(1 - x)}^{2}}$
Let $S = 1 + 3 x + 5 x^{2} + 7 x^{3} + 9 x^{4} + . . . . . . .$
$x S = x + 3 x^{2} + 5 x^{3} + 7 x^{4} + 9 x^{5} + . . . . . . .$
$S - x S = 1 + (3 x - x) + (5 x^{2} - 3 x^{2}) + (7 x^{3} - 5 x^{3}) + (9 x^{4} - 7 x^{4}) + . . . . . . .$
$S (1 - x) = 1 + 2 x (1 + x + x^{2} + x^{3} + . . . .)$
$= 1 + 2 x \times \frac{1}{1 - x}$
$= \frac{x + 1}{1 - x}$
$S = \frac{x + 1}{(1 - x)^{2}}$

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Sum 1+3x+5x2+7x3+9x4+..... to infinity is (x<1)

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Sum $1 + 3 x + 5 x^{2} + 7 x^{3} + 9 x^{4} + . . . . .$ to infinity is $(x < 1)$