If $| x | < 1$ , then find the sum of the series $2 + 4 x + 6 x^{2} + 8 x^{3} + . . .$

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Solution

Let $S = 2 + 4 x + 6 x^{2} + 8 x^{3} + . . . .$

$\Rightarrow x S = 2 x + 4 x^{2} + 6 x^{3} + . . . .$

From these equations, we get,

$\Rightarrow S (1 - x) = 2 + 2 x + 2 x^{2} + 2 x^{3} + . . .$

$= 2 (1 + x + x^{2} + . . . .)$

$\Rightarrow 1 + x + x^{2} + . . .$ is an infinite GP with $a = 1, r = x$ and $| r | = | x | < 1$

Therefore, sum of the series = $\frac{1}{1 - x}$

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

$\Rightarrow s = \frac{2}{(1 - x)^{2}}$

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