SECT. 9-1 SEQUENCES

Sequence

An ordered collection of numbers in a prescribed order defined by a function f(n)

The values, an are called terms

a1, a2, a3, a4 ,..., an ,...

notation {a1, a2, a3, a4, ...} {an} {an}n1

1) Write the first five terms of the sequence

a)

b)

an nn 1

an 1 n (n 1)

3n

Defining Sequences

Determine the pattern in the sequenceAnd use pattern to determine the nth term using inductive reasoning

1, 24

, 79

, 1416

, 2325

...

12 212 ,

22 222 ,

32 232 ,

42 242 ,

52 252 ...

an n2 2n2

2) Write the next two terms for the given sequence

72

, 4, 92

, 5 ....

rewrite : 3.5, 4, 4.5, 5, 5.5, 6, ...

A recursively defined sequence: given the first term, all other terms are defined using that term. d1 = 3.5 and {dn}= dn+1

3) Find next three terms: pattern?

5, 10, 20, 40, ...

4) Find the general term an

35

, - 425

, 5125

, - 6625

, 73125

...

Numerator: start with 1 for first term, add 2

Denominator: powers of 5

Notice terms alternate signs n1 )1(or )1( n

5) Write and expression for the nth term

2, 1, 45

, 57

, 69

,...

rewrite21

, 33

, 45

, 57

, 69

,...

an n 1

2n 1 there may be several ways to write the nth term

Convergence and Divergence of a sequence

We say that a sequence converges to a limit L if

If no limit exists then an diverges.If the terms increase without bound, {an} diverges to infinity

limn

an L

Convergence or Divergence?

Convergence and Divergence ?

limn

an L

Properties of Sequences

if limn

an L and limn bn K then

1. limn

an bn LK

2. limn can cL

3. limn

anbn Lk

4. limn

anbn

LK

, bn 0 and K 0

6) Evaluate

limn

1 n

n

limn

1n

1

Squeeze Theorem for Sequences:If {an}, {bn}, and {cn} are sequences and an ≤ bn ≤ cn for every n andif , then lim limn nn n

a L c

lim nnb L

7) Determine whether the sequence converges or diverges. If it converges, find the limit.

a.

b.

c.

2

5n

ne

1( 1)n

232 n

diverges

to 0converges

to 2converges

8) Determine whether the sequence converges or diverges. If it converges, find the limit.

a.

b.

c.

2cos3nn

1 1( 1)nn

to 0converges

to 0converges

!

2lim

21

!1

nnn

n

n

to 0converges

9) Determine whether the sequence converges or diverges. If it converges, find the limit

2

2

2

2 53n

lim 53

nnn

nnnan

Factorial (!)

5! = 54 321

10) simplify

25!23!

11) simplify

n 2 !n!

Increasing and Decreasing Sequences

•A sequence {an} is increasing if

an an1 for all n 1 a1 a2 a3 ...

•A sequence {an} is decreasing if

an an1 for all n 1 a1 a2 a3 ...

•A sequence {an} is monotonic if it is either always increasing or always decreasing

Monotonic? Sequences

Bounded Sequences

•A sequence {an} is bounded from above if

there is a number M such that an M for all n

•A sequence {an} is bounded from below if

there is a number M such that an M for all n

Bounded Sequences

•A sequence {an} is called Bounded if it is bounded either from above or below. •If a sequence {an} is bounded

and monotonic,then it converges.

12) Is the sequence {an} bounded?

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