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

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Solution

Let the sum of given series be $S$
$S = 1 + (1 + b) r + (1 + b + b^{2}) r^{2} + (1 + b + b^{2} + b^{3}) r^{3} . . . . .$
Multiplying both sides by $b r$ , we get
$S b r = 0 + b r + (b^{2} + b) r^{2} + (b + b^{2} + b^{3}) r^{3} + (1 + b + b^{2} + b^{3}) b r^{4} . . . . . . . . . . . . . . . . . .$
Subtracting both equations, we get
$S (1 - b r) = 1 + r + r^{2} + r^{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \infty$

$\Rightarrow S (1 - b r) = \frac{1}{1 - r}$
$\Rightarrow S = \frac{1}{(1 - r) (1 - b r)}$

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Find the sum of the infinite series1+(1+b)r+(1+b+b2)r2+(1+b+b2+b3)r3+....,r and b being proper fractions.

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