PreCalculus Class Notes SS6 Summation Notation
Summation notation is used to write series efficiently. The symbol Σ, sigma, indicates the sum.
n
∑a
k =1
k
= a1 + a2 + a3 + " + an
The letter k is called the index of summation. The numbers 1 and n represent the subscripts of the
first and last term in the series. They are called the lower limit and upper limit of the summation,
respectively.
Example
Evaluate each series.
5
∑k2
k =1
4
∑5
k =1
Example
Write the series using summation notation. Let the lower limit equal 1.
1 1 1 1 1 1 1
+ + + + + +
23 33 43 53 63 73 83
Example
Write the series using summation notation. Let the lower limit equal 1.
1 2 3 4 5 6
+ + + + +
2 3 4 5 6 7
6
∑ ( 2k − 5 )
k =3
Rewriting summations with an index that starts at 1
11
15
∑ 3k
∑
k =5
Let n be the new index. When k = 5, then
n = 1 so n = k – 4 and also k = n + 4,
now substitute
11− 4
∑
n =5− 4
k
k =7
Let n be the new index. When k = 7, then
n = 1 so n = k – 6and also k = n + 6,
now substitute
7
15− 6
n =1
n =7 −6
3 ( n + 4 ) = ∑ ( 3n + 12 )
∑
9
n+6 =∑ n+6
n =1
Example
Rewrite each summation so that the index starts with n = 1.
7
∑k2
k =4
30
∑ ( 2k − 3 )
k =8
Properties for Summation Notation
Let a1, a2, a3, …, an and b1, b2, b3, …, bn be sequences, and c be a constant.
n
∑ ca
Multiply by a constant
k =1
n
k
= c∑ ak
k =1
n
n
n
k =1
k =1
k =1
n
n
n
k =1
k =1
k =1
∑ ( ak + bk ) = ∑ ak + ∑ bk
Add terms
∑ ( ak − bk ) = ∑ ak − ∑ bk
Subtract terms
Formulas for Simple Sums
n
∑ c = nc
Sum of a constant
k =1
n ( n + 1)
2
k =1
n
n ( n + 1)( 2n + 1)
k2 =
∑
6
k =1
n
∑k =
Sum of the index
(arithmetic series)
Sum of the index squared
Example
Use properties for summation notation to find each sum.
5
6
∑3
k =1
40
∑k
k =1
∑k
k =1
22
2
7
∑k
∑ 2k
k =1
k =1
∑ ( 2k
14
k =1
2
− 3)
2