Find sum of $n$ terms of following series ?
$1 + 3 x + 5 x^{2} + 7 x^{3} + \dots$

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Solution

$S = 1 + 3 x + {5 x}^{2} + {7 x}^{3} + . . . . . . + (2 n + 1) x^{n}$
$x S = x + {3 x}^{2} + {5 x}^{3} + . . . . . . + (2 n - 1) x^{n} + {(2 n + 1)}^{n + 1} x - ----------------------------------------------------------- -$
$\Rightarrow S (1 - x) = 1 + 2 x + {2 x}^{2} + {2 x}^{3} + . . . . . . + {2 x}^{n} + (2 n + 1) x^{n + 1}$
$= [1 + (2 n + 1) x^{n + 1}] + 2 x (1 + x + x^{2} + . . . . . . x^{n})$
$\Rightarrow S = {[1 + (2 n + 1) x^{n + 1}] + \frac{2 x (x^{n} - 1)}{(x - 1)}} \frac{1}{(1 - x)}$

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