Summation notation is a shorthand way of writing certain expressions that are
sums of terms. As an example, consider the sum of the integers from 1 to 5:
1 + 2 + 3 + 4 + 5
This expression can be thought of in the following way: Suppose we have some
quantity i that initially has the value 1 and then takes on successive values of 2, 3,
4, and 5. The expression is the sum of i at all its different values. The summation
notation is
∑
5
i=1
i
The uppercase Greek letter sigma, Σ, denotes summation. Here the number 1 is
the lower limit of summation, and the number 5 is the upper limit of summation.
The variable i is called the index of summation. The index of summation is initially set equal to the lower limit and then keeps increasing its value by 1 until it
reaches the upper limit. All the values that the index of summation takes on are
added together. Thus
∑
5
i=1
i = 1 + 2 + 3 + 4 + 5 = 15
Similarly
∑
3
i=1
i = 1 + 2 + 3 = 6
And
∑
8
i=4
i = 4 + 5 + 6 + 7 + 8 = 30
In these examples, the expression after the summation sign is just i, the index of summation. However, what appears after the summation sign can be any
expression, and the successive values of the index are simply substituted into the
expression. Thus
∑
5
i=1
i
2
= 1
2
+ 2
2
+ 3
2
+ 4
2
+ 5
2
= 55
Summation and Product Notation
B B
A p p e n d i x
805