document.write( "Question 982376: What is the sum of the infinite geometric series 3/4-9/16+27/64-81/256+ ..?
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Algebra.Com's Answer #603312 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Instead of doing your problem for you, I will do one EXACTLY IN EVERY\r\n" );
document.write( "DETAIL like yours, so that when you do yours, you can use this as a\r\n" );
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document.write( "I will find the sum of this infinite geometric series instead:\r\n" );
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document.write( "\"2%2F3-4%2F9%2B8%2F27-16%2F81%2B%22%22-%22%22%2A%22%22%2A%22%22%2A%22%22\"\r\n" );
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document.write( "We should first check to see if it really is a geometric series, for sometimes\r\n" );
document.write( "books contain errors/typos.  If it is a geometric series then this must be true:\r\n" );
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document.write( "\"matrix%281%2C2%2C2nd%2Cterm%29%2Fmatrix%281%2C2%2C1st%2Cterm%29\"\"%22%22=%22%22\"\"matrix%281%2C2%2C3rd%2Cterm%29%2Fmatrix%281%2C2%2C2nd%2Cterm%29\"\"%22%22=%22%22\"\"matrix%281%2C2%2C4th%2Cterm%29%2Fmatrix%281%2C2%2C3rd%2Cterm%29\"\r\n" );
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document.write( "So we calculate the three ratio to see if they are all equal:\r\n" );
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document.write( "\"matrix%281%2C2%2C2nd%2Cterm%29%2Fmatrix%281%2C2%2C1st%2Cterm%29\"\"%22%22=%22%22\"\"%28-4%2F9%29%2F%282%2F3%29\"\"%22%22=%22%22\"\"expr%28-4%2F9%29%2Aexpr%283%2F2%29\"\"%22%22=%22%22\"\"-12%2F18\"\"%22%22=%22%22\"\"-2%2F3\"\r\n" );
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document.write( "\"matrix%281%2C2%2C3rd%2Cterm%29%2Fmatrix%281%2C2%2C2nd%2Cterm%29\"\"%22%22=%22%22\"\"%288%2F27%29%2F%28-4%2F9%29\"\"%22%22=%22%22\"\"expr%288%2F27%29%2Aexpr%28-9%2F4%29\"\"%22%22=%22%22\"\"-72%2F108\"\"%22%22=%22%22\"\"-2%2F3\"\r\n" );
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document.write( "\"matrix%281%2C2%2C4th%2Cterm%29%2Fmatrix%281%2C2%2C3rd%2Cterm%29\"\"%22%22=%22%22\"\"%28-16%2F81%29%2F%288%2F27%29\"\"%22%22=%22%22\"\"expr%28-16%2F81%29%2Aexpr%2827%2F16%29\"\"%22%22=%22%22\"\"-432%2F1296\"\"%22%22=%22%22\"\"-2%2F3\"\r\n" );
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document.write( "They are all the same so it is a geometric series with common ratio = \"-2%2F3\"\r\n" );
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document.write( "It is a geometric series with a defined sum when taken to infinitely many\r\n" );
document.write( "terms because the ratio \"r=-2%2F3\" is greater than -1 and less than +1.\r\n" );
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document.write( "\"S%5Binfinity%5D\"\"%22%22=%22%22\"\"a%5B1%5D%2F%281-r%29\", where a1 = 1st term = \"2%2F3\"\r\n" );
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document.write( "\"S%5Binfinity%5D\"\"%22%22=%22%22\"\"%282%2F3%29%2F%281-%28-2%2F3%29%29\"\"%22%22=%22%22\"\"%282%2F3%29%2F%281%2B2%2F3%29\"\"%22%22=%22%22\"\"%282%2F3%29%2F%283%2F3%2B2%2F3%29\"\"%22%22=%22%22\"\"%282%2F3%29%2F%285%2F3%29\"\"%22%22=%22%22\"\"expr%282%2F3%29expr%283%2F5%29\"\"%22%22=%22%22\"\"6%2F15=2%2F5\"\r\n" );
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document.write( "Answer to this problem = \"2%2F5\"\r\n" );
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document.write( "Now do your problem EXACTLY the same way.\r\n" );
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document.write( "Edwin
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