If a , x are real numbers, | a | 1 , | x | 1 , then 1 + ( 1 + a ) x + ( 1 + a + a 2 ) x 2 + ? is equal to

Question

If $a, x$ are real numbers, $| a |$ $< 1$ , $| x |$ $< 1$ , then $1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots \infty$ is equal to

Smart Achievers · Accepted Answer

(3)

$1 + (1 + a) x + (1 + a + a^{2}) x^{2} + \dots to \infty$

$= \frac{1}{1 - a} [(1 - a) + (1 - a^{2}) x + (1 - a^{3}) x^{2} + \dots to \infty]$

$= \frac{1}{1 - a} [(1 + x + x^{2} + \dots to \infty) - (a + a^{2} x + a^{3} x^{2} + \dots to \infty)]$

$= \frac{1}{1 - a} [\frac{1}{1 - x} - \frac{a}{1 - a x}] = \frac{1}{(1 - x) (1 - a x)}$

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