Evaluate $1 + 2 x + 3 x^{2} + 4 x^{3} + . . .$ upto infinity, where $| x | < 1.$

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Solution

Let $S = 1 + 2 x + 3 x^{2} + 4 x^{3} + . . .$ (i)
$x S = x + 2 x^{2} + 3 x^{3} + . . .$ (ii)
(i)-(ii) $\Rightarrow$ $(1 - x) S = 1 + x + x^{2} + x^{3} + . . .$
or $S = \frac{1}{{(1 - x)}^{2}}$

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