Evaluate: $1 + (1 + b) r + (1 + b + b^{2}) r^{2} + . . .$ to infinite terms for $| b r | < 1.$

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Solution

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

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1 + (1 + b) r + (1 + b + b^{2}) r^{2} + (1 + b + b^{2} + b^{3}) r^{3} + . . . ., r

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