document.write( "Question 381236: Express the sum below using summation notation. Use 1 as the lower limits of summation and i for the index of summation.\r
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Algebra.Com's Answer #270431 by Edwin McCravy(20055) $\"\"$ $\"About$

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document.write( "Since it's not terribly long, let's write the entire series out:\r\n" );
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document.write( "That's a sum of fractions.  Every one of those fractions has a\r\n" );
document.write( "numerator and a denominator.\r\n" );
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document.write( "Therefore we want to get this series in a short-hand form like this\r\n" );
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document.write( "We are told that the starting number of the index \"i\" is 1. So\r\n" );
document.write( "let's put \"1\" in place of the words \"starting integer\"\r\n" );
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document.write( "Now let's make a list of all the values of the integer \"i\", the list\r\n" );
document.write( "of all the numerators and the list of all the denominators:\r\n" );
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document.write( " i    numerators   denominators\r\n" );
document.write( " 1       1             3 \r\n" );
document.write( " 2       2             4\r\n" );
document.write( " 3       3             5\r\n" );
document.write( " 4       4             6\r\n" );
document.write( " 5       5             7 \r\n" );
document.write( " 6       6             8  \r\n" );
document.write( " 7       7             9\r\n" );
document.write( " 8       8            10 \r\n" );
document.write( " 9       9            11\r\n" );
document.write( "10      10            12\r\n" );
document.write( "11      11            13\r\n" );
document.write( "12      12            14 \r\n" );
document.write( "13      13            15\r\n" );
document.write( "14      14            16\r\n" );
document.write( "15      15            17\r\n" );
document.write( "16      16            18\r\n" );
document.write( "17      17            19\r\n" );
document.write( "18      18            20\r\n" );
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document.write( "Notice that there are 18 terms and therefore the ending integer for\r\n" );
document.write( "the index \"i\" is 18.  Therefore in place of the words \"ending_integer\",\r\n" );
document.write( "we can put \"18\":\r\n" );
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document.write( "Look at the list of values for the index, that is, the \"i\"'s. Then look at\r\n" );
document.write( "the list of numerators. Notice that they are all exactly the same as the\r\n" );
document.write( "\"i\"'s. Therefore the numerators are all the same as i.  Therefore we can put\r\n" );
document.write( "\"i\" in place of the word \"numerator\" and we have this:\r\n" );
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document.write( "Now look at the list of denominators.  Notice that every one of them is\r\n" );
document.write( "2 more than the corresponding value in the list of index values \"i\" and\r\n" );
document.write( "the numerators.  So we see that each denominator is the value of both the \r\n" );
document.write( "index \"i\" and the numerator with 2 added to it.  Therefore we can put \"i+2\"\r\n" );
document.write( "in place of the word \"denominator\".  So the final answer is\r\n" );
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document.write( "Edwin

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