Download B. So, what is an infinite sequence?

Lesson: Infinite Sequences and Their Limits
A. Introduction
This is a lesson on Infinite Sequences. Why do we need to study Infinite Sequences? They are
the foundation for Infinite Series, a major topic of AP Calculus BC. Infinite Series is covered in
chapter 9 of our textbook. I consider this topic the most interesting in calculus. I hope you find it
interesting as well.
What is an infinite series? It is an infinite summation. What does it mean for an infinite number
of numbers to be added up? Does the sum add up to some finite number? Or does it converge, or
does it diverge? If it converges, what number does it converge to? How do you find and infinite
sum? Is this actually possible?
We will answer these questions on our next lesson, that is, Infinite Series. This lesson is on
infinite sequences, but we’ll touch on Infinite Series here a bit.
Let’s do this by looking at some examples.
Example 1. Suppose we have the summation (geometric series) 1+
1 1 1 1
+ + + + ...
2 4 8 16
What
do you think this infinite sum adds up to?
Example 2.
1 1 1 1
1+ + + + + ... (harmonic series). What do you think this one adds up to?
2 3 4 5
Example 3. 1-
Example 4,
1 1 1 1
+ - + - ...
2 3 4 5
1 1 1 1
+ + + + ...
12 2 2 32 4 2
The answer to this is really surprising!
We will learn all about this amazing topic in, later in the next lesson. We first need to study
Infinite Sequences. Now, let’s begin our topic for this lesson, Infinite Sequences.
B. So, what is an infinite sequence?
A Sequence is simply a list of numbers in a special order. Or you could say, “A sequence is an
ordered set of numbers. More specifically, a sequence is a function whose domain is a subset of
the positive integers. For example, the Fibonacci numbers, {1, 1, 2, 3, 5, 8, 13,¼} , are a
sequence, where the next number in the list is the sum of the previous two.
1
Another sequence is {1, 4,9,16,...} , you can see that the next number is the next perfect square in
the sequence.
So how do you symbolize a sequence, so that you don’t write so much? The convention for a
sequence is, an , where {a1,a2 ,a3 ,a4 ,...,an ,...} ; So, this is the range, and the domain is, n = 1, 2,
3, … Therefore a sequence is denoted by curly brackets. So, the last example is an = n 2 , so this
means, {1, 4,9,16,...}
Let’s look at some more examples.
1 1 1 1
Example 5: 1, , , , ,..., an ,... Each of the numbers listed is called a term. So, the first term
2 4 8 16
is 1, the second term is ½, and so on, and an is called the nth term.
1
ì1 ü
Using the two common notations for this sequence; you would write an = , or í ý .
n
în þ
n
Example 6: Given the sequence, an =
to find the first 5 terms, you simply plug in 1,2,3, for
n +1
1 2 3 4 5
n, and you’d get, , , , , ,... .
2 3 4 5 6
Before we continue on limits, you should have read pages 435 to 440. Arithmetic and
Geometric Sequences are left for you to study. These are not covered in the video.
Now, stop the video and you try the following example.
Example 7:
(-1)n
an = 2
Find the first 6 and 100th term of this sequence.
n +1
Okay, how did you do? Easy right?
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C. The limit of a sequence
How do you find the limit of an infinite sequence? Let’s do some examples to show this.
Example 8: Determine whether the given sequence converges or diverges, and if it converges,
find the limit.
2n -1
an =
n
The sequence converges and its limit is 2. Notice that we used the properties of page 438 to find
this limit.
Example 9:
2
lim(3 + (-1)n )
x®¥
The limit does not exist.
Example 10:
1ö
æ
lim ç 5 - 2 ÷
x®¥ è
n ø
It converges and the limit is 5.
Examples 11-12:
æ 1ö
lim ç ÷
x®¥ è n ø
æ 1ö
lim nsin ç ÷ Þ
Þ
x®¥
1
è nø
n
Here the limit is 1
æ 1ö
lim sin ç ÷
x®¥
è nø
The limit is 0.
Example 13:
Determine whether the sequence with the given nth term converges or diverges. If it converges,
find the limit.
n -1
an = (-1)n
, n = 1,2, 3,...
n
D. Theorem 2 The Sandwich Theorem for Sequences
ì cos n ü
Example 14: Show that the sequence í
ý converges, and find the limit.
î n þ
Because cos x £ 1 for all x (negative or positive), then you can rewrite the expression as
cosn cos n 1
1 cosn 1
£
£ for all integers ³ 1. Therefore you can rewrite it as, - £
£ .
n
n
n
n
n
n
1
1
come from? If n < 0 , you get - .
n
n
cos n
1
1
= 0 , because lim æç - ö÷ = lim æç ö÷ = 0
Therefore the sequence converges, since lim
x®¥
x®¥ è
n
n ø x®¥ è n ø
Now, where does the -
E. Theorem 3 Absolute Value Theorem
Consider the sequence {an } . If lim an = 0, thenlim an = 0
x®¥
x®¥
3