document.write( "Question 1043793: x+y +(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+...to infinite terms\r
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Algebra.Com's Answer #659036 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "   x and y are real numbers such that |x| < 1  and |y|< 1.\r\n" );
document.write( "   Find the infinite sum  \r\n" );
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document.write( "Let \"S\" be the infinite sum \r\n" );
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document.write( "S = .\r\n" );
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document.write( "Multiply S by (x-y). Then\r\n" );
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document.write( "S = .\r\n" );
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document.write( "Now notice that\r\n" );
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document.write( " = ,               (1)\r\n" );
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document.write( " = ,        (2)\r\n" );
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document.write( " =   (3)   (make yourself this calc . . . )\r\n" );
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document.write( "And so on . . . \r\n" );
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document.write( "So, I suppose (and I am almost sure) that each parenthesed term in the original sum, multiplied by (x-y) will give .   (4)\r\n" );
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document.write( "   //\"The margins of this page are too narrow . . . \"\r\n" );
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document.write( "Thus we have\r\n" );
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document.write( "  S*(x-y) =  -  =     (5)\r\n" );
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document.write( "     Now apply the formula for the infinite sum of a geometric progression\r\n" );
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document.write( "= .\r\n" );
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document.write( "Simplify it and then cancel the factor (x-y) in both sides.\r\n" );
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document.write( "Finally, you will get\r\n" );
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document.write( "    S =  = 1 - .\r\n" );
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document.write( "Again, the key is the idea with the formulas (1), (2), (3), (4), (5).\r\n" );
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document.write( "Answer.  S =  = 1 - .\r\n" );
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