document.write( "Question 117354: Find the sum of the series
\n" ); document.write( "1 + 2x + 3x^2 + 4x^3 ...\r
\n" ); document.write( "\n" ); document.write( "I cannot understand how to find the ratio\r
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\n" ); document.write( "1 + 2x + 3x^2 + 4x^3 ...
\n" ); document.write( "--\/---\/-------\/--------
\n" ); document.write( "--2x---1.5x---1.33x\r
\n" ); document.write( "\n" ); document.write( "Now what? \n" ); document.write( "
Algebra.Com's Answer #86116 by Edwin McCravy(20077)\"\" \"About 
You can put this solution on YOUR website!
Find the sum of the series
\n" ); document.write( "1 + 2x + 3x^2 + 4x^3 ...\r
\n" ); document.write( "\n" ); document.write( "I cannot understand how to find the ratio\r
\n" ); document.write( "\n" ); document.write( "Like
\n" ); document.write( "1 + 2x + 3x^2 + 4x^3 ...
\n" ); document.write( "--\/---\/-------\/--------
\n" ); document.write( "--2x---1.5x---1.33x\r
\n" ); document.write( "\n" ); document.write( "Now what?
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document.write( "Since the ratios are not equal, you have\r\n" );
document.write( "shown this series is not a geometric series.\r\n" );
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document.write( "1 + 2x + 3x2 + 4x3 ...\r\n" );
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document.write( "I hope you are taking calculus, for ordinary\r\n" );
document.write( "algebra cannot handle this sort of problem.\r\n" );
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document.write( "If we take the antiderivative term by term,\r\n" );
document.write( "we get\r\n" );
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document.write( "x + x2 + x3 + x4 + ｷｷｷ + C\r\n" );
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document.write( "and this, all except for the arbitrary constant C IS \r\n" );
document.write( "a geometric series with a1 = x and r = x\r\n" );
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document.write( "Now we use the formula for the sum of an infinite\r\n" );
document.write( "geometric series:\r\n" );
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document.write( "       a1\r\n" );
document.write( "S･ = 覧覧予r\n" );
document.write( "      1-r\r\n" );
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document.write( "       x\r\n" );
document.write( "S･ = 覧覧予r\n" );
document.write( "      1-x\r\n" );
document.write( "                               x\r\n" );
document.write( "x + x2 + x3 + x4 + ｷｷｷ + C = 覧覧予r\n" );
document.write( "                              1-x\r\n" );
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document.write( "So now we take the derivative term by term\r\n" );
document.write( "and get:\r\n" );
document.write( "                                (1-x)(1) - x(-1)\r\n" );
document.write( "1 + 2x + 3x2 + 4x3 + ｷｷｷ + 0 = 覧覧覧覧覧覧覧覧覧\r\n" );
document.write( "                                    (1-x)2\r\n" );
document.write( "   1 - x + x\r\n" );
document.write( "= 覧覧覧覧覧予r\n" );
document.write( "     (1-x)2  \r\n" );
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document.write( "     1\r\n" );
document.write( "= 覧覧覧覧\r\n" );
document.write( "   (1-x)2\r\n" );
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document.write( "Edwin
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