201-NYB-05 Handout 3
Summation Notation
Consider the following sums.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
1 1 1 1
1+ + + + ... + n
2 4 8 2
1 1 1
+ + ... + + ...
2 3 n
P
The symbol , Sigma (the Greek letter for S), can be used to express the sums above.
In general, summation notation is presented in the form
n
X
ai
i=1
where ai describes the terms to be added, and i is called the index. Each term is evaluated at i, then all values
are added up, beginning with the value when i = 1 and ending with the value when i = n. For example,
7
X
ai =
i=1
Note that the index is used to keep track of the terms to be added. We can use any letter we like for the index.
Typically, mathematicians use i, j, k, m, and n for indices.
Example 1. Write the following sums using summation notation.
1 1 1 1
a) 1 + + + + =
4 9 16 25
1 1 1
b) + + ··· + =
1+n 2+n 2n
1
Example 2. Expand each sum.
4
X
a) (3i2 + 2)
i=1
n
X i 1
b) ln · =
n n
i=1
Properties of Sigma Notation
Let a1 , a2 , . . . , an and b1 , b2 , . . . , bn represent two sequences of terms and let c be a constant. The following
properties hold for all positive integers n and for integers m, with 1 ≤ m ≤ n.
n
X
(1) c=
i=1
n
X
(2) cai =
i=1
n
X
(3) (ai + bi ) =
i=1
n
X
(4) (ai − bi ) =
i=1
n
X
(5) ai =
i=1
2
Summation Formulas
n
X
(1) i=
i=1
n
X
(2) i2 = 12 + 22 + 32 + . . . + n2 =
i=1
n
X
(3) i3 = 13 + 23 + 33 + . . . + n3 =
i=1
Example 3. Evaluate the following sums.
100
X
a) (4 + 3i) =
i=1
n
X
b) (i − 3)2 =
i=1
n
X
c) (j 3 − j 2 ) =
j=1
