<\/span><\/h3>\nTo handle the absolute value inside sigma, we need to consider two cases: when the argument inside the absolute value is positive and when it is negative.<\/p>\n
When the argument inside the absolute value is positive, we do not need to modify the expression. We can simply continue evaluating the sigma as usual.<\/p>\n
However, when the argument inside the absolute value is negative, we can rewrite it as a positive value. This is because the absolute value of a negative number is equal to its positive counterpart. By doing so, we can simplify the expression and make it easier to work with.<\/p>\n
Let’s illustrate these two cases with an example:<\/p>\n
Consider the following sigma expression:<\/p>\n
[ sum_{i=1}^{5} | x_i – 3 | ]<\/p>\n
To evaluate this expression, we need to examine the value of ( x_i – 3 ) for each term from ( i = 1 ) to ( i = 5 ).<\/p>\n
If ( x_i – 3 ) is positive, we leave it unchanged. For example, if ( x_1 = 7 ), then ( x_1 – 3 = 4 ), which is positive. Thus, the first term becomes ( |4| = 4 ).<\/p>\n
On the other hand, if ( x_i – 3 ) is negative, we rewrite it as a positive value. Let’s say ( x_2 = 1 ), then ( x_2 – 3 = -2 ), which is negative. So, the second term becomes ( |-(-2)| = 2 ).<\/p>\n
By applying this process to each term, we can find the values for all the absolute values inside the sigma and then proceed to sum them up as usual.<\/p>\n
Therefore, the expression<\/p>\n
[ sum_{i=1}^{5} | x_i – 3 | ]<\/p>\n
can be simplified to<\/p>\n
[ | x_1 – 3 | + | x_2 – 3 | + | x_3 – 3 | + | x_4 – 3 | + | x_5 – 3 | ]<\/p>\n
where each absolute value is evaluated based on whether ( x_i – 3 ) is positive or negative.<\/p>\n