Sum to $i n f i n i t e$ terms the following series:
$1 + 3 x + 5 x^{2} + 7 x^{3} + . . . ., | x | < 1$ .

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Solution

$S = 1 + 3 x + 5 x^{2} + 7 x^{3} + . . .$ $. . . . . . . . . . . (1)$
$S x = x + 3 x^{2} + 5 x^{3} + . . . .$ $. . . . . . . . . . . . . . . (2)$

$e q^{n} (1) - e q^{n} (2)$ , we get
$(1 - x) S = 1 + 2 x + 2 x^{2} + 2 x^{3} + \dots . . .$ $. . . . . . . . . (3)$

This is a $G . P$ . of common ratio $= x$

So, using sum of $G . P . = \frac{a}{(1 - r)}$
Here, $a = f i r s t t e r m$ and $r = c o m m o n r a t i o$

So,
$(1 - x) S = 1 + \frac{2 x}{(1 - x)}$

$(1 - x) S = \frac{1 + x}{(1 - x)}$

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

Hence, this is the answer.

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