# ) n f ```Math 151 Section 6.1
Summation Notation
Definition: A sequence of real numbers is a function,
range the real numbers. We usually write
Example:
f (n) 
1
 an
n
f (n ) , with domain the nonnegative integers and
f ( n )  an .
We often write the sequence as an ordered list, listing enough terms to see
the formula and then … . For this example, we could write the sequence as
1 1 1
1, , , , ...
2 3 4
When we want to add many terms of a given sequence, we use summation notation. The Greek letter
sigma, ∑, is used for summations.
5 1
1 1 1 1
Example: 1      
2 3 4 5 n 1 n
n
is called the index in the sum.
Examples: Evaluate each sum.
9
1.
n
n4
4
2.

1
k
 1  
k  2
Reindexing: If the index
2
 1 
 

k

1

k  2
8
7

k varies from 2 to 8 then k  1  j varies from 1 to 7. We can write,
1
2
j 1 j
.
All the usual rules of arithmetic apply to finite sums.
5 1
5
3


7
k

3

7



k
k
k


k 2
k 2
k 2
5
Example:
Two formulas for adding certain sums:
n ( n  1)
k 
2
k 1
n
n
2
k 
k 1
n ( n  1)( 2 n  1)
6
Examples:
4
1.
 i ( i  3)
i 1
 5  2 k  3k
100
2.
k 0
2

 1
1 



 2
k  12 
k 1 k
10
3.
Example 3 is called a telescoping or collapsing sum.
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