Sigma Notation

1. Sigma/Summation
Notation
5.2

2. Consider the following sum:
12 + 22 + 32 + 42 + 52
Each of the terms is
in the form of k2,
where k is an integer
This can be written from 1 to 5.
in sigma notation
as:
5
å=
1
2
k
k

3. Sigma Notation
n
n
i a a a a + + + = å=
i
... 1 2
1
 i -> the index of summation
 ai -> the ith term
 i, n -> lower and upper bounds of
summation

4. Determine the sum
å=
+
4
1
2
k
k = (1+ 2) + (2 + 2) + (3+ 2) + (4 + 2)
= 3+ 4 + 5 + 6
=18

5. Determine the sum
5
å=
3
3
k
k = 3(3) + 3(4) + 3(5)
= 9 +12 +15
= 36

6. Determine the sum
å=
- +
4
0
( 1) (2 1)
k
k k
= (-1)0 (2(0) +1) + (-1)1(2(1) +1) + (-1)2 (2(2) +1) + (-1)3 (2(3) +1) + (-1)4 (2(4) +1)
=1- 3+ 5 - 7 + 9
= 5

7. Determine the sum
4
å=
1
sin( )
k
pk
= sin(1p ) + sin(2p ) + sin(3p ) + sin(4p )
= 0 + 0 + 0 + 0
= 0

8. Summation Properties
å å
= =
ca =
c a
k 1 1
å å å
= = =
n
k
k
n
k
± = ±
n
k
k
n
k
k
n
k
k k a b a b
1 1 1
n
å==
k
c nc
1

9. Useful Theorems
( )
2
+ = + + + + = å=
1 2 3 ... 1
1
k n n n
n
k
( )( )
2 2 2 2 2 + + = + + + + = å=
1 2 3 ... 1 2 1
6
1
k n n n n
n
k
( )
4
2 2
3 3 3 3 3 + = + + + + = å= k n n n
1 2 3 ... 1
1
n
k

10. Determine the sum
12
å=
1
2 2
i
i
ù
úû
12
2 2
i
= é + +
2 12(12 1)(2(12) 1)
êë
6
2 12(13)(25)
= é
=1300
å==
1
i
ù
úû
êë
6

11. Determine the sum
å=
6
( +
) 1
2 1
i
i
å å
= =
6
i
úû
6(7)(13) ù
+ 6
= é
êë
= 97
= +
6
1
6
1
2 1
i i

12. Determine the sum
( å=
+
) 6
1
2 2
i
i
å å å
= = =
4(6)
i i
= + +
6(7)(13) + 4 é 6(úû
úû
7)
ù
= 6
2
êë
ù
êë= é
=199
6
1
6
1
6
1
2 4 4
i i i

13. Where to get more information
If you didn´t understand press here
Or visit
http://www.youtube.com/watch?v=hEPk36Yncxg