If $x < 1$ , sum the series $1 + 2 x + 3 x^{2} + 4 x^{3} + . . . .$ upto infinity.

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Solution

Given that $x < 1$
Let $S = 1 + 2 x + 3 x^{2} + 4 x^{3} . . . . . . . . . \infty$
Multiplying both sides by $x$ , we get
$S x = 0 + x + 2 x^{2} + 3 x^{3} . . . . . . . . \infty$
Subtract both eqns, we get
$S (1 - x) = 1 + x + x^{2} + x^{3} . . . . . . . . \infty$
Clearly, this is an infinite series with common ratio $x < 1$
$\Rightarrow S (1 - x) = \frac{1}{1 - x}$
$\Rightarrow S = \frac{1}{(1 - x)^{2}}$

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