we have
x = \dfrac{1}{1-a}
or (1-a) = \dfrac{1}{x} or a = 1 –
\dfrac{1}{x} = \dfrac{x-1}{x}
similarly
b = \dfrac{y-1}{y}
now as |a | and |b| both <
1 so |ab| < 1 and hence
1+ ab + a^2b^2 + a^3b^3 + \cdots= \dfrac{1}{1-ab} = \dfrac{1}{1- \frac{x-1}{x} * \frac{y-1}{y}}
= \dfrac{xy}{xy - (x - 1)(y – 1)}
= \dfrac{xy}{x + y – 1}
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