A series is a summation of the terms in a sequence. For the sequence the series can be written as If the terms are given by a rule, it's usually more compact to write it using sigma notation. The sequence above is described by the explicit rule The series can be written as Here, by substituting with integers (under ) through (above ) the individual terms of the series are obtained, and the sigma itself indicates that between each term is a plus sign. This alternative notation is useful if the series contains a large number of terms, and especially useful when the sequence is infinite. Then, the infinity symbol is written above the sigma.
Now, there are fives, and the sum is Here, is the mean of the first and last term, which means that it can be written as This is multiplied by which is the number of terms, leading to the formula
Determine the sum of all positive integers between and