If |x| < 1, |y| < 1 and x $\ne$ y, then the sum to infinity
of the following series
(x + y) + (x2+xy+y2) + (x3+x2y + xy2+y3) + ....
Solution
(x + y) + (x<sup>2</sup>+xy+y<sup>2</sup>) + (x<sup>3</sup>+x<sup>2</sup>y + xy<sup>2</sup>+y<sup>3</sup>) + ....
<br><br>By multiplying and dividing x – y :
<br><br>$${{\left( {{x^2} - {y^2}} \right) + \left( {{x^3} - {y^3}} \right) + \left( {{x^4} - {y^4}} \right) + ...} \over {x - y}}$$
<br><br>= $${{\left( {{x^2} + {x^3} + {x^4} + ....} \right) - \left( {{y^2} + {y^3} + {y^4} + ...} \right)} \over {x - y}}$$
<br><br>= ${{{{{x^2}} \over {1 - x}} - {{{y^2}} \over {1 - y}}} \over {x - y}}$
<br><br>= $${{\left( {{x^2} - {y^2}} \right) - xy\left( {x - y} \right)} \over {\left( {1 - x} \right)\left( {1 - y} \right)\left( {x - y} \right)}}$$
<br><br>= ${{x + y - xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}$
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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