Before we generalize the indexed notation, let us give some examples with the sigma form.
A notation ∑ _{i=1}^{5}x_{ i} means that i is replaced by whole numbers starting with 1 until the number 5 is reached. Thus
and
Hence, the notation ∑ _{i=1}^{n} tells us:
to add the numbers x_{i},
to start with i = 1, that is, with x_{1},
to stop with i = n, that is, with x_{n} (where n is some positive integer).
As an example, let us assign the following values: x_{1} = 10, x_{2} = 8, x_{3} = 2, x_{4} = 15, and x_{5} = 22. Then we have:
The name i is a dummy variable, any other name could be used. We could have used j, the expression would have be exactly the same, hence:
Now let us find ∑ _{i=1}^{4}3x_{ i} based on the previous values. Again we start with i = 1 and we replace 3x_{i} with its value:
Similarly, let use find ∑ _{i=2}^{5}(x_{ i} - 8): This is:
One should be careful with the parentheses. The expression ∑ _{i=2}^{5}(x_{ i} - 8) is not the same as ∑ _{i=2}^{5}x_{ i} - 8. The later evaluates to (the 8 is not included in the sum):
We also use sigma notation in the following way:
The same principle applies here. j is replaced in the expression j^{2} by numbers starting with 1 and ending with 4, and then adding up all four terms.
For the sigma notation we have three important transformation rules:
Rule 1: if c is a constant, then:
Rule 2: if c is a constant, then:
Rule 3: Adding the term can be distributed:
Proof:
∑ _{i=1}^{n}cx_{ i} = cx_{1} + cx_{2} + … + cx_{n-1} + cx_{n} | |||
= c ⋅ (x_{1} + x_{2} + … + x_{n-1} + x_{n}) = c ⋅∑ _{i=1}^{n}x_{ i} | |||
∑ _{i=1}^{n}c = _{n-times} = n × c = nc | |||
∑ _{i=1}^{n}(x_{ i} + y_{i}) = (x_{1} + y_{1}) + … + (x_{n} + y_{n}) | |||
= (x_{1} + … + x_{n}) + (y_{1} + … + y_{n}) = ∑ _{i=1}^{n}x_{ i} + ∑ _{i=1}^{n}y_{ i} |
End of proof