Notes 9.4 – Sequences and Series

I. Sequences

A.) A progression of numbers in a pattern.

1.) FINITE – A set number of terms

2.) INFINITE – Continues forever

2, 5, 8, 11,... 242

1 1 11, , , ,...

2 4 8

B.) Notation - ak where a is the sequence and k refers to the kth term.

C.) Explicitly defined sequence– Allows us to substitute k into an equation to find the kth term.

D.) Recursively defined sequence– Defines each term by using the previous term.

E.) Ex. 1 - Define the following sequence both explicitly and recursively.

first + diff( -1)ka k

4 6( -1)

2 6ka k

k

1 4a

1 6 for 2k ka a k

Exp:

Rec:

4, 10, 16, 22,...

II. Types of Sequences

A.) Arithmetic Sequence – any sequence with a common difference between terms.

B.) Geometric Sequence - any sequence with a common ratio between terms.

2, 5, 8, 11,...

1 1 11, , , ,...

2 4 8

C.) Ex. 2- Define the following sequence both explicitly and recursively.

( -1)first ratio kka

13

844

k

ka

1 84a

1

3for 2

4k ka a k

Exp. Rec.

189 56784, 63, , ,...

4 16a

D.) Constructing a Sequence – Ex. 3 - The third and sixth terms of a sequence are -12 and 48 respectively.

Find an explicit formula for the sequence if it is

1.) arithmetic and 2.) geometric

1.) -

112 2a d

148 5a d

1 12 2a d

12 2 48 5d d

20d

1 48 5a d

1 52a

52 20( 1)ka k

2.)

2112 a r

5148 a r

34 r 1 23

12

4a

1

32

3

124

4

k

ka

23

112 4a

3 4r

3312 4

k

ka

Divide the two equations

III. Fibonacci Sequence

A.) Summation of successive terms. Can only be defined recursively

B.) Ex. 4– ak = 1, 1, 2, 3, 5, 8, 13,…

1

2

2 1

1

1

for 3k k k

a

a

a a a k

IV. Summation Notation

A.) The way we express the sum of a sequence of n terms {a1, a2, a3, …, an}

“the sum of ak from k = 1 to n”

k = the index of summation.

1

n

kk

a

B.) Ex. 5-

C.) Ex. 6-

5

1

2k

k

2 4 6 8 10 30 12

0

cos( )k

k

cos(0) cos( ) cos(2 )

cos(3 )... cos(12 ) 1

V. Sum of a Finite Arith. Seq.

A.)

1 21

...n

k nk

a a a a

1

2na a

n

12 12

na n d

B.) Proof -

11

2n

k nk

a n a a

1 1 1 11

2 ... ( 1)n

kk

a a a d a d a n d

2 ... ( 1)n n n na a d a d a n d

1 1 1 11

2 ...n

k n n n nk

a a a a a a a a a

1

1 2

nn

kk

a aa n

1 ( 1)na a n d 11

2 ( 1)2

n

kk

na a n d

1 1

1

( 1)

2

n

kk

a a n da n

C.) Ex. 7- Find the sum of the following sequences:

A.) 3, 6, 9, 12,…, 21 B.) 111, 108, 105,…27

B.) 27 111 / 3 1 29

111 2729

2

2001

1A.) 2

3 217

2

84

na an

VI. Sum of a Finite Geom. Seq.

A.) 1 2

1

...n

k nk

a a a a

1 1

1

na r

r

B.) Ex. 8- Find the sum of the following sequence:

3, 6, 12, 24,…,48, 96

C.) Ex. 9- Find the sum of the following sequence:

10, 20, 30, 40,…

1

1

96 3 2

32 2

6

n

n

n

6

1

3 1 2

1 2

189

n

kk

a

No Sum!!!

Divergent Sequence

D.) Ex. 9 - Find the sum of the following sequence:

0.1, 0.01, 0.001, 0.0001, …

Convergent Sequence - It has a sum!

1

0.11111111...

10.1

9

n

kk

a

VII. Sum of a Infinite Geom. Seq.A.) A.K.A. an INFINITE SERIES

B.) Not a true sum. This is considered a Partial Sum

If it converges…

1 21

... ...k nk

a a a a

1 21

lim lim ...n

k nn n

k

a a a a s

If | r |< 1

C.) Ex. 10- Find the sum of the following series:

1

1 1 1k

kk k

aa a r

r

2

1 .72

.320

3

1

1

2 .7k

k