Section  11.1     MATH  1152Q  Part  1   Sequences   Notes  

Page 1 of 4

A sequence is an ordered collection of objects. Example: A sequence of musical notes Example: You may have seen a pattern recognition “IQ” puzzle similar to this:

In calculus, a sequence can be thought of as a list of numbers indexed by the natural numbers 1,2,3,4, … :

, , , , … , , … or a function from the natural numbers to the real numbers. The number is called the first term, is the second term, and in general is the th term. The sequence is also denoted by

or Note that doesn’t have to start at . Example: 1, 3, 5, 7, 9, … is a sequence of odd natural numbers. A sequence can be finite or infinite, but for this class we’re mostly interested in infinite sequences and what happens as you look further down the sequence. Some sequences can be defined by giving an explicit formula for the th term. For example, the Geometric Sequence (See Sec 11.1 Example 11):

. Each subsequent number is determined by multiplying the previous term by a fixed, nonzero number. Task 1 Example: Write the first four terms of the sequence where . Task 2 Example: Find a formula for the general term for the sequence of odd natural numbers. Then find a formula for the sequence { 1, -3, 5, -7, 9, .... }.

1a 2a 3a 4a na

1a 2a na n

{ }1 2 3, , , a a a

{ }na { } 1n na ¥

=

n 1

n

11n

na a r -=

1{ }n na ¥=

22 3 1na n n= - +

na

Section  11.1     MATH  1152Q  Part  1   Sequences   Notes  

Page 2 of 4

Example: Find a formula for the general term of the sequence

assuming that the pattern of the first few terms continues. Example: 1,1,2,3,5,8,13, … the Fibonacci Sequence can be defined by:

, and for all . Each subsequent number is the sum of the previous two.

Task 3 Example: Consider the sequence . (Create a table with Desmos.com)

It appears that the terms of the sequence are approaching as becomes large. In

fact, the difference

can be made as small as we like by taking sufficiently large. We indicate this by writing

In general, the notation

means that the terms of the sequence approach as becomes large. Task 4 Definition (copy from page 696) A sequence has the limit and we write

if we can make the terms as close to as we like by taking sufficiently large. If

exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

na3 4 5 6 7, , , , , 5 25 125 625 3125ì ü- -í ýî þ

1 1a = 2 1a = 2 1n n na a a+ += + 1n ³

1nnan

=+

1nnan

=+

1 n

111 1nn n

- =+ +

n

lim 11n

nn®¥

=+

lim nna L

®¥=

{ }na L n

{ }na L

na L n lim nna

®¥

Section  11.1     MATH  1152Q  Part  1   Sequences   Notes  

Page 3 of 4

Task 5: Determine whether the sequence where is convergent. (Ex.4 pg698)

Task 6 Theorem (from book, Theorem 3 on pg 697)

If and when is an integer, then .

Why?

Task 7 Ex: Determine whether the sequence where is convergent. (Ex. 6 pg 698)

Theorem If and are convergent sequences and is a constant, then

•   .

•   .

•   .

•   provided that .

{ }na 10nnan

=+

( )limx

f x L®¥

= ( ) nf n a= n

{ }naln

nnan

=

{ }na { }nb c

( )lim n nna b

®¥+ lim limn nn n

a b®¥ ®¥

= +

lim nnca

®¥lim nnc a

®¥=

( )lim n nna b

®¥lim limn nn na b

®¥ ®¥= ×

lim n

nn

ab®¥

lim

limnn

nn

a

b®¥

®¥

= lim 0nnb

®¥¹

Section  11.1     MATH  1152Q  Part  1   Sequences   Notes  

Page 4 of 4

Theorem (Squeeze Theorem for sequences) If for and , then .

Task 8: Determine whether the sequence {𝑏"} where 𝑏" =(%&)(

" is convergent. (Ex. 8 pg699).

Task 9 Theorem (read carefully, from book, Theorem 7 on page 699) If and the function is continuous at , then

.

Task 10: Determine whether the sequence where is convergent. (Ex. 9 pg 699).

Task 11: Re-read Thm 3 and Thm 7 from the book. What are the differences between the two?

n n na b c£ £ n N³ lim limn nn na c L

®¥ ®¥= = lim nn

b L®¥

=

lim nna L

®¥= f L

( ) ( ) ( )lim limn nn nf a f a f L

®¥ ®¥= =

{ }na sinna npæ ö= ç ÷è ø