Summation and Product Notation
807
Rules of Summation
1. ∑
q
i=p
(a i + b i ) = ∑
q
i=p
a i + ∑
q
i=p
b i
2. ∑
q
i=p
(a i − b i ) = ∑
q
i=p
a i − ∑
q
i=p
b i
3. ∑
q
i=p
ca i = c ∑
q
i=p
a i
where c is a constant
To prove rule 1, note that
a p + b p + a p+1 + b p+1 + c + a q + b q
= a p + a p+1 + c + a q + b p + b p+1 + c + b q
because of the commutative property of addition. The proof of rule 2 is similar.
To prove rule 3, note that
ca p + ca p+1 + c + ca q = c(a p + a p+1 + c + a q )
because of the distributive property. This rule allows a constant to be “moved
through” a summation.
Sometimes a summation can be represented by an even shorter expression
that does not involve adding separate terms. For example, according to Exercise 7
of Section 2.2,
∑
n
i=1
i
2
=
n(n + 1)(2n + 1)
6
(1)
so that the value of ∑
5
i=1
i
2
can be found by substituting the upper limit, 5, into the
right side of (1), giving
5(5 + 1)(2 * 5 + 1)
6
= 55
as before. Section 2.2 and its exercises give a number of other “closed form” expressions for certain summations, all of them provable by mathematical induction.
Product notation is a shorthand way of writing certain expressions that are
products of factors. Product notation is very similar to summation notation, except
it uses the uppercase Greek letter pi, Π and the various items are multiplied rather
than added. There is an index of multiplication, a lower limit, and an upper limit.
For example,
q
7
i=3
i = (3)(4)(5)(6)(7) = 2520