Find the sum to $n$ terms of the series
$1 + 2 x + 3 x^{2} + 4 x^{3} + . . .$

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Solution

$S = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots \dots .$ upto n terms

$\to S = 1 + 2 x + 3 x^{2} + 4 x^{3} + \dots \dots . + n x^{n - 1}$

$x s = x + 2 x^{2} + 3 x^{3} + \dots \dots . + n x^{n}$

$S - x s = 1 + x + x^{2} + x^{3} + \dots \dots x^{n - 1} - n x^{n}$

This is a $G . P$ and $S_{G . P} = (\frac{1 - x^{n}}{1 - x})$

So, $s - x s = (\frac{1 - x^{n}}{1 - x}) - n x^{n}$

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

$S \Rightarrow \frac{(1 - x^{n})}{{(1 - x)}^{2}} - \frac{n x^{n}}{(1 - x)}$

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