Find the sum of the infinite series $1 + (1 + a) b + (1 + a + a^{2}) b^{2} + (1 + a + a^{2} + a^{3}) b^{3} + . . . . b$ and a being proper fractions.

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Solution

$1 + (1 + a) b + (1 + a + a^{2}) b^{2} + (1 + a + a^{2} + a^{3}) b^{3} + . . . = (1 + b + b^{2} + b^{3} + . . .) + (b + b^{2} + b^{3} + . . .) a + (b^{2} + b^{3} + b^{4} . . .) a^{2} + . . . = (1 + b + b^{2} + b^{3} + . . .) + (1 + b + b^{2} + b^{3} + . . .) a b + (1 + b + b^{2} + b^{3} + . . .) a^{2} b^{2} + . . . = (1 + b + b^{2} + b^{3} + . . .) (1 + a b + a^{2} b^{2} + . . .)$
$= \frac{1}{1 - b} \times \frac{1}{1 - a b} = \frac{1}{(1 - b) (1 - a b)}$

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