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Question Number 45283 by peter frank last updated on 11/Oct/18

if x = 1 + a + a^{2} + a^{3} + \dots \dots \dots

y = 1 + b + b^{2} + b^{3} + \dots \dots \dots

prove that

1 + ab + a^{2} b^{2} + a^{3} b^{3} + \dots \dots \dots = \frac{xy}{x + y - 1}

Answered by ajfour last updated on 11/Oct/18

x = \frac{1}{1 - a}; y = \frac{1}{1 - b}

z = 1 + a b + a^{2} b^{2} + a^{3} b^{3} + \dots

= \frac{1}{1 - a b} = \frac{1}{1 - (1 - \frac{1}{x}) (1 - \frac{1}{y})}

= \frac{x y}{x y - (x - 1) (y - 1)}

= \frac{x y}{x + y - 1} .

Commented by peter frank last updated on 11/Oct/18

thank you sir

Commented by maxmathsup by imad last updated on 11/Oct/18

t h i s i s t r u e i f w e k n o w t h a t ∣ a ∣< 1 a n d ∣ b ∣< 1 \dots