SOLUTION: x+y +(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+...to infinite terms

Question 1043793: x+y +(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+...to infinite terms

Answer by ikleyn(52790) (Show Source): You can put this solution on YOUR website!
.
Let me reformulate the problem in this way:

   x and y are real numbers such that |x| < 1  and |y|< 1.
   Find the infinite sum

Solution

Let "S" be the infinite sum 

S = .

Multiply S by (x-y). Then

S = .

Now notice that

 = ,               (1)

 = ,        (2)

 =   (3)   (make yourself this calc . . . )

And so on . . . 

So, I suppose (and I am almost sure) that each parenthesed term in the original sum, multiplied by (x-y) will give .   (4)

   //"The margins of this page are too narrow . . . "

Thus we have

  S*(x-y) =  -  =     (5)

     Now apply the formula for the infinite sum of a geometric progression

= .

Simplify it and then cancel the factor (x-y) in both sides.

Finally, you will get


    S =  = 1 - .


Again, the key is the idea with the formulas (1), (2), (3), (4), (5).

Answer.  S =  = 1 - .

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