If a = sum_{n=0}^infty x^n,b = sum_{n=0}^inft | vedclass

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(C) Given $a = \sum_{n=0}^\infty x^n = \frac{1}{1-x}$,which implies $x = \frac{a-1}{a}$.Similarly,$b = \sum_{n=0}^\infty y^n = \frac{1}{1-y}$,which im

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$ac + bc = ab + c$

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(C) Given $a = \sum_{n=0}^\infty x^n = \frac{1}{1-x}$,which implies $x = \frac{a-1}{a}$.Similarly,$b = \sum_{n=0}^\infty y^n = \frac{1}{1-y}$,which implies $y = \frac{b-1}{b}$.Also,$c = \sum_{n=0}^\infty (xy)^n = \frac{1}{1-xy}$,which implies $xy = \frac{c-1}{c}$.Substituting $x$ and $y$ in the expression for $xy$:$\left(\frac{a-1}{a}\right) \left(\frac{b-1}{b}\right) = \frac{c-1}{c}$$\frac{ab - a - b + 1}{ab} = \frac{c-1}{c}$$c(ab - a - b + 1) = ab(c-1)$$abc - ac - bc + c = abc - ab$$-ac - bc + c = -ab$$ab + c = ac + bc$.

If $a = \sum_{n=0}^\infty x^n$,$b = \sum_{n=0}^\infty y^n$,and $c = \sum_{n=0}^\infty (xy)^n$,where $|x|, |y| < 1$,then which of the following is true?

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