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