Algebra.Com's Answer #756379 by ikleyn(52792)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find the sum of the series, if it converges. Otherwise, enter DNE.
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document.write( "sum from n to infinity of: 3/(n(n+2))
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document.write( " The problem formulation, as it is presented in the post, is INCORRECT.\r
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document.write( " The correct formulation, after my editing, is below:\r
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document.write( " Find the sum of the series { sum over n from 1 to infinity of  }.\r\n" );
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document.write( "Solution\r
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document.write( "Use the identity\r\n" );
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document.write( "  =  -  \r\n" );
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document.write( "which you can easily prove by writing the right hand side with the common denominator.\r\n" );
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document.write( "It will give you\r\n" );
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document.write( " { sum from n to infinity of } = \r\n" );
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document.write( " =  -  +\r\n" );
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document.write( " +  -  +\r\n" );
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document.write( " + -  +\r\n" );
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document.write( " + -  +\r\n" );
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document.write( " + -  +\r\n" );
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document.write( " + . . . and so on . . . \r\n" );
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document.write( "From this writing, yo can see that in the (infinite) sum all the terms cancel each other, and only two very first terms survive.\r\n" );
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document.write( "These terms are + , so the series does converge and the sum from 1 to infinity is equal to \r\n" );
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document.write( " + =  + =  =   = 2.25. ANSWER\r\n" );
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document.write( "CHECK.\r\n" );
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document.write( "n Sum\r\n" );
document.write( " of the terms\r\n" );
document.write( " in column 2\r\n" );
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document.write( "1 1.0000 1.0000\r\n" );
document.write( "2 0.3750 1.3750\r\n" );
document.write( "3 0.2000 1.5750\r\n" );
document.write( "4 0.1250 1.7000\r\n" );
document.write( "5 0.0857 1.7857\r\n" );
document.write( "6 0.0625 1.8482\r\n" );
document.write( "7 0.0476 1.8958\r\n" );
document.write( "8 0.0375 1.9333\r\n" );
document.write( "9 0.0303 1.9636\r\n" );
document.write( "10 0.0250 1.9886\r\n" );
document.write( "11 0.0210 2.0096\r\n" );
document.write( "12 0.0179 2.0275\r\n" );
document.write( "13 0.0154 2.0429\r\n" );
document.write( "14 0.0134 2.0563\r\n" );
document.write( "15 0.0118 2.0680\r\n" );
document.write( "16 0.0104 2.0784\r\n" );
document.write( "17 0.0093 2.0877\r\n" );
document.write( "18 0.0083 2.0961\r\n" );
document.write( "19 0.0075 2.1036\r\n" );
document.write( "20 0.0068 2.1104\r\n" );
document.write( "21 0.0062 2.1166\r\n" );
document.write( "22 0.0057 2.1223\r\n" );
document.write( "23 0.0052 2.1275\r\n" );
document.write( "24 0.0048 2.1323\r\n" );
document.write( "25 0.0044 2.1368\r\n" );
document.write( "26 0.0041 2.1409\r\n" );
document.write( "27 0.0038 2.1447\r\n" );
document.write( "28 0.0036 2.1483\r\n" );
document.write( "29 0.0033 2.1516\r\n" );
document.write( "30 0.0031 2.1547\r\n" );
document.write( "31 0.0029 2.1577\r\n" );
document.write( "32 0.0028 2.1604\r\n" );
document.write( "33 0.0026 2.1630\r\n" );
document.write( "34 0.0025 2.1655\r\n" );
document.write( "35 0.0023 2.1678\r\n" );
document.write( "36 0.0022 2.1700\r\n" );
document.write( "37 0.0021 2.1721\r\n" );
document.write( "38 0.0020 2.1740\r\n" );
document.write( "39 0.0019 2.1759\r\n" );
document.write( "40 0.0018 2.1777\r\n" );
document.write( "41 0.0017 2.1794\r\n" );
document.write( "42 0.0016 2.1810\r\n" );
document.write( "43 0.0016 2.1826\r\n" );
document.write( "44 0.0015 2.1841\r\n" );
document.write( "45 0.0014 2.1855\r\n" );
document.write( "46 0.0014 2.1868\r\n" );
document.write( "47 0.0013 2.1881\r\n" );
document.write( "48 0.0013 2.1894\r\n" );
document.write( "49 0.0012 2.1906\r\n" );
document.write( "50 0.0012 2.1917\r\n" );
document.write( "51 0.0011 2.1929\r\n" );
document.write( "52 0.0011 2.1939\r\n" );
document.write( "53 0.0010 2.1949\r\n" );
document.write( "54 0.0010 2.1959\r\n" );
document.write( "55 0.0010 2.1969\r\n" );
document.write( "56 0.0009 2.1978\r\n" );
document.write( "57 0.0009 2.1987\r\n" );
document.write( "58 0.0009 2.1996\r\n" );
document.write( "59 0.0008 2.2004\r\n" );
document.write( "60 0.0008 2.2012\r\n" );
document.write( "61 0.0008 2.2020\r\n" );
document.write( "62 0.0008 2.2028\r\n" );
document.write( "63 0.0007 2.2035\r\n" );
document.write( "64 0.0007 2.2042\r\n" );
document.write( "65 0.0007 2.2049\r\n" );
document.write( "66 0.0007 2.2056\r\n" );
document.write( "67 0.0006 2.2062\r\n" );
document.write( "68 0.0006 2.2068\r\n" );
document.write( "69 0.0006 2.2074\r\n" );
document.write( "70 0.0006 2.2080\r\n" );
document.write( "71 0.0006 2.2086\r\n" );
document.write( "72 0.0006 2.2092\r\n" );
document.write( "73 0.0005 2.2097\r\n" );
document.write( "74 0.0005 2.2103\r\n" );
document.write( "75 0.0005 2.2108\r\n" );
document.write( "76 0.0005 2.2113\r\n" );
document.write( "77 0.0005 2.2118\r\n" );
document.write( "78 0.0005 2.2123\r\n" );
document.write( "79 0.0005 2.2127\r\n" );
document.write( "80 0.0005 2.2132\r\n" );
document.write( "81 0.0004 2.2136\r\n" );
document.write( "82 0.0004 2.2141\r\n" );
document.write( "83 0.0004 2.2145\r\n" );
document.write( "84 0.0004 2.2149\r\n" );
document.write( "85 0.0004 2.2153\r\n" );
document.write( "86 0.0004 2.2157\r\n" );
document.write( "87 0.0004 2.2161\r\n" );
document.write( "88 0.0004 2.2165\r\n" );
document.write( "89 0.0004 2.2168\r\n" );
document.write( "90 0.0004 2.2172\r\n" );
document.write( "91 0.0004 2.2176\r\n" );
document.write( "92 0.0003 2.2179\r\n" );
document.write( "93 0.0003 2.2183\r\n" );
document.write( "94 0.0003 2.2186\r\n" );
document.write( "95 0.0003 2.2189\r\n" );
document.write( "96 0.0003 2.2192\r\n" );
document.write( "97 0.0003 2.2195\r\n" );
document.write( "98 0.0003 2.2198\r\n" );
document.write( "99 0.0003 2.2201\r\n" );
document.write( "100 0.0003 2.2204\r\n" );
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document.write( "Solved.\r
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document.write( "This trick, which I used in my solution, is very well known.\r
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document.write( "See the lesson\r
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document.write( " - Calculations with fractions\r
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document.write( "in this site.\r
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