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Let \( x=1+a+a^{2}+\ldots \ldots \ldots . \). to \( \infty \) and \( y=1+b+b^{2}+\ldots \ldots \ldots \). to \( \infty \), where \( |a|1 \) and \( |b|1 \).
Prove that \( : 1+a b+a^{2} b^{2}+\ldots \ldots \cdots \cdots=\frac{x y}{x+y-1} \).
\( \mathrm{P} \)
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