If a = ∑ n = 0 ∞ x n , b = ∑ n = 0 ∞ y n , c = ∑ n = 0 ∞ ( x y ) n where ∣ x ∣ , ∣ y ∣

Question

If a = ∑ n = 0 ∞ ​ x n , b = ∑ n = 0 ∞ ​ y n , c = ∑ n = 0 ∞ ​ ( x y ) n where ∣ x ∣ , ∣ y ∣ < 1 , then

TestHub · Accepted Answer

a = 1 − x 1 ​ ⇒ x = a a − 1 ​ … (i); b = 1 − y 1 ​ ⇒ y = b b − 1 ​ … (ii); c = 1 − x y 1 ​ . c = 1 − ab ( a − 1 ) ( b − 1 ) ​ 1 ​ = a + b − 1 ab ​ ⇒ a c + b c − c = ab ⇒ a c + b c = ab + c .

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