MAT 1236Calculus III
Appendix E
Sigma Notations
+ Maple Lab
http://myhome.spu.edu/lauw
HW and…
No WebAssign HW Download Appendix E HW Quiz on Thursday: 11.1 II, App. E, 11.2 I No class tomorrow. Take the time to
prepare for the next exam HW key posted (?) for the last problem in
HW
Summation Notation
Ambiguity
302 2 2 2 2
1
1 2 3 30k
k
Ambiguity
30
1
2 1 3 2 4 30 32
1 ! 2 1! 3 2! 31 30!k
k k
k k
Review:Sigma Notation (Summation)
nmZnmaaaaa nmmm
n
mii
,, 21
Final value (upper limit)
Initial value (lower limit)Index
Example 0
42 2 2 2 2
1
42 2 2 2 2
1
4
1
1 2 3 4
1 2 3 4
1 1 1 1 1 4
i
j
k
i
j
Indices are “dummy”
Summation
Q: Can you name one place in Calculus II where we use the summation notation?
Theorem
n n
i ii m i m
n n n
i i i ii m i m i m
c a c a
a b a b
Finite limits
Common Formulas
6
)12)(1(321
2
)1(321
2222
1
2
1
nnnni
nnni
n
i
n
i
Example 1 (Telescoping Sum)
n
i
ii1
331
3 33 3
1 1 1
1 1n n n
i i i
i i i i
Index Shifting
Sigma representation of a summation is not unique
22226
2
22
22224
0
22
22225
1
22
543211
543211
54321
i
i
i
i
i
i
Index Shifting
Sigma representation of a summation is not unique
22226
2
22
22224
0
22
22225
1
22
543211
543211
54321
i
i
i
i
i
i
Index Shifting
Sigma representation of a summation is not unique
22226
2
22
22224
0
22
22225
1
22
543211
543211
54321
i
i
i
i
i
i
Index Shifting
Sigma representation of a summation is not unique
22226
2
22
22224
0
22
22225
1
22
543211
543211
54321
i
i
i
i
i
i
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
decrease the index by 1
increase the i in the summation by 1
Index Shifting Rules
6
2
24
0
25
1
2 11iii
iii
increase the index by 1
decrease the i in the summation by 1
Example
7 6 8
3 2 4
1 1
1 2k k k
k k k
k k k
Example 2
Rewrite
such that the lower limit is 0
n
k
kk1
2 1
Example 1 Revisit
n
i
ii1
331
Using “+…+” is ambiguous. We would like to avoid these kind of notations.
3 33 3
1 1 1
1 1n n n
i i i
i i i i
Expectations
Avoid “+…+” by using summations Break up the summations before
canceling.
Maple Lab 11.2
Explore and understand the convergence of a series some examples of standard series
Definition
Given a series
We define the partial sum sequence {Sn}
1 2 31
kk
a a a a
1 2 31
n
n n kk
S a a a a a
Example 1
1
1
k k
1S
Series Partial Sum Sequence {Sn}
1
1
2k
k
nS
nS
3S 2S
1S 3S 2S