Section 8.1: Sequences

Practice HW from Stewart Textbook (not to hand in)

p. 565 # 3-33 odd

Sequences

Sequences are collection of numbers or objects

that is ordered by the positive integers.

Notation: (known as the

terms of the sequence).naaaaa ,, , , , 4321

Example 1: Write the first five terms of the

sequence .

Solution:

1n

nan

Example 2: Write the first five terms of the

sequence .

Solution:

n

n

na3

)1(

Describing the nth term of a sequence

Involves writing a formula describing the pattern

the sequence of numbers follows.

Example 3: Find a formula for the general term

of the sequence .

Solution:

},15 ,11 ,7 ,3{ na

Example 4: Find a formula for the general term

of the sequence .

Solution:

},25

4 ,

16

3 ,

9

2 ,

4

1{ na

Convergence of Sequences

Sequences whose behavior approaches a limiting

value are said to converge to this value

Example 5: What value does the sequence

appear to converge to?

Solution:

n2

1

Definition

A sequence has the limit L and we write

which is read approaches L as n approaches

if we can make the terms as close to L as we

like by taking n sufficiently large. If exists,

then the sequence is convergent. Otherwise, it is

divergent.

Lann

lim

na

na

nn

a

lim

Useful Facts for Showing Convergence of

Sequences

1. L Hopital’s Rule for indeterminate forms .

2. Absolute Value Theorem: If , then

.

3. Using Maple to graph.

0lim na

0lim na

Example 6: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution:

nnan 2

Example 7: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution:

13

12

2

n

nan

Example 8: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution:

12

n

n

ne

ea

Example 9: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution:

nan cos

Example 10: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution: (In typewritten notes)

)ln()12ln( nnan

Example 11: Determine whether the sequence

converges or diverges. If the

sequence converges, find the limit.

Solution:

1)1(

2

n

na nn

Factorial

Recall that , read as n factorial, is defined to

be

Note: .

! n

123)3)(2)(1( ! nnnnn

1! 0

Example 12: Compute .

Solution:

! 5

Example 13: Determine whether the sequence

converges or diverges. If the sequence

converges, find the limit.

Solution: (In typewritten notes)

! n

ea

n

n