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If a=1,+x+x^2... and b=1+y+y^2+...[x|lt1...

If `a=1,+x+x^2...` and `b=1+y+y^2+...[x|lt1and]y|lt1`, then prove that `1-xy+x^2y^2+x^3y^3+....=(ab)/(a+b-1)`

Text Solution

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`a=1+x+x^2+...=1/(1-x)implies(1-x)=1 implies a-1=ax implies=(a-1)/a similarly,y=(b-1)/b`
`therefore1+xy+x^2y^2+....=1/(1-xy)=1/((a-1)(b-1))/1-(ab)=(ab)/(ab-(ab-a-b+1))=(ab)/(a+b-1)`
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