If -x-1 and -y-1- find the sum of infinity of the following series- x-y-x2-x y-y2-x3-x- 2 y-x y- 2-y- 3-

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) + ( ...

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Byju's Answer

Standard XII

Mathematics

Special Integrals - 2

If |x|<1 and ...

Question

If |x|<1 and |y|<1, find the sum of infinity of the following series: (x+y) + (x²+xy+y²) + (x³+x^2y+xy^2+y^3) + ....

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Solution

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

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

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

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

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

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

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

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

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