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Question

If |x|<1 and |y|<1, find the sum of infinity of the following series: (x+y) + (x​​​​​​2+xy+y​​​​​​2) + (x​​​​​​3+x^2y+xy^2+y^3) + ....

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Solution

Let Sn = [(x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + .............n terms ]

Multiply and divide with ( x - y) then we obtain

Sn =( x - y) [(x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + .............n terms ] / ( x - y)

Sn =( x - y)(x + y) + ( x - y)(x2 + xy + y2) + ( x - y)(x3 + x2y + xy2 + y3) + .............n terms ] / ( x - y)

Sn = (x2 - y2) + (x3 - y3) + (x4 - y4) + .............n terms ] / ( x - y)

Sn = [(x2 + x3 + x4 + -----------n terms) - (y2 + y3 + y4 + -----------n terms)] / ( x - y).

Sn = [x2( 1 + x + x2 + -----------n terms) - y2 (1 + y + y2 + -----------n terms)] / ( x - y).

Sn = [{x2( 1 - xn ) / ( 1 - x)} - { y2 (1 - yn) / (1 - y)}] / ( x - y).

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