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Question

If $open vertical bar a close vertical bar less than 1$ and $open vertical bar b close vertical bar less than 1$ , then the sum of the series $1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis$ is

$\frac{1}{(1 - a) (1 - b)}$
$\frac{1}{(1 - a) (1 - a b)}$
$\frac{1}{(1 - b) (1 - a b)}$
$\frac{1}{(1 - a) (1 - b) (1 - a b)}$

The correct answer is: $fraction numerator 1 over denominator open parentheses 1 minus b close parentheses left parenthesis 1 minus a b right parenthesis end fraction$

We have,
$1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis infinity blank$
$equals not stretchy sum from n equals 1 to infinity of left parenthesis 1 plus a plus a to the power of 2 end exponent plus horizontal ellipsis plus a to the power of n minus 1 end exponent right parenthesis b to the power of n minus 1 end exponent$
$equals not stretchy sum from n equals 1 to infinity of open parentheses fraction numerator 1 minus a to the power of n end exponent over denominator 1 minus a end fraction close parentheses b to the power of n minus 1 end exponent$
$equals not stretchy sum from n equals 1 to infinity of fraction numerator b to the power of n minus 1 end exponent over denominator 1 minus a end fraction minus not stretchy sum from n equals 1 to infinity of fraction numerator a to the power of n end exponent b to the power of n minus 1 end exponent over denominator 1 minus a end fraction$
$equals fraction numerator 1 over denominator 1 minus a end fraction not stretchy sum from n equals 1 to infinity of b to the power of n minus 1 end exponent minus fraction numerator a over denominator 1 minus a end fraction not stretchy sum from n equals 1 to infinity of open parentheses a b close parentheses to the power of n minus 1 end exponent$
$equals fraction numerator 1 over denominator 1 minus a end fraction open square brackets 1 plus b plus b to the power of 2 end exponent plus horizontal ellipsis infinity close square brackets minus fraction numerator a over denominator 1 minus a end fraction left square bracket 1 plus a b plus open parentheses a b close parentheses to the power of 2 end exponent plus horizontal ellipsis infinity right square bracket$
$equals fraction numerator 1 over denominator 1 minus a end fraction cross times fraction numerator 1 over denominator 1 minus b end fraction minus fraction numerator a over denominator open parentheses 1 minus a close parentheses left parenthesis 1 minus a b right parenthesis end fraction$
$equals fraction numerator 1 over denominator open parentheses 1 minus a b close parentheses left parenthesis 1 minus b right parenthesis end fraction$

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