LESSON 18 SEQUENCES
Notation: The symbol is used to denote the set of positive integers. Thus, .
Definition A sequence is a function whose domain is a set of the form , where N is a positive integer.
Up to this point, we have worked with functions of a real variable. That is, the domain of the function was either the set of real numbers or was a subset of the set of real numbers.
Example Consider the sequence .
The domain of this sequence is , and we have that
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.
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If f is a sequence, then for each positive integer n in the domain of the sequence, there corresponds a real number in the range of the sequence. If the domain of the sequence is the set , where N is a positive integer , then the numbers in the range of the sequence may be listed by writing
, , , ...... , , ......
The number is called the first term of the sequence, is called the second term of the sequence, is called the third term of the sequence, is called the fourth term of the sequence, and is called the n th term of the sequence. Of course if , then the numbers in the range of the sequence would be
, , , ...... , , ......
and the number is called the first term of the sequence, is called the second term of the sequence, is called the third term of the sequence, is called the fourth term of the sequence, and is called the n th term of the sequence.
COMMENT: This is the reason that some people define a sequence to be a function whose domain is the set of positive integers . Thus, by their definition, the function would not be a sequence because it is not defined at . By our definition, f is a sequence whose domain is the set .
NOTATION: When working with a sequence, it is customary to use a subscript notation. Thus, instead of writing , we write . Thus, for our example above, instead of having , we would have . Then we would have written the following
instead of
instead of
instead of
instead of
instead of
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COMMENT: The most common names for functions of a real variable are f, g, h, and k. The most common names for sequences are a, b, and c.
Examples Find the domain of the following sequences. Then find the first four terms of the sequence.
1.
The domain of this sequence is the set = .
, , , and
Answer: ; 0, 1, ,
NOTE: is the first term of the sequence
is the second term of the sequence
is the third term of the sequence
is the fourth term of the sequence
is the n th term of the sequence
2.
The domain of this sequence is the set = .
, , , and
Answer: ; 5, , 1,
3.
The domain of this sequence is the set = .
, , , and
Answer: ; , , ,
NOTE: is the first term of the sequence
is the second term of the sequence
is the third term of the sequence
is the fourth term of the sequence
is the n th term of the sequence
4.
NOTE: Sometimes, sequences are given this way.
The domain of this sequence is the set = .
The first term of the sequence is .
The second term of the sequence is .
The third term of the sequence is .
The fourth term of the sequence is .
Answer:; , , , or
; , , ,
Definition A sequence has the limit L, written , if for every , there exists a positive integer N such that for all .
If a sequence has a limit, then we say that the sequence converges. If a sequence does not have a limit, then we say that the sequence diverges.
We have the following limit properties for sequences.
Theorem (Properties of Limits for Sequences) If and are sequences and and , and c is any real number, then
1. =
2. = +
3. =
4. =
5. = , provided that
Proof Will be provided later.
Theorem (The Sandwich Theorem for Sequences) Given the sequences , , and . If and and if for all , where N is a positive integer, then .
Proof Will be provided later.
Since sequences are not differentiable, then L’Hopital’s Rule does not hold for sequences. However, we have the following theorem which will allow us is to apply L’Hopital’s Rule.
Theorem Let be a sequence with a domain of . Let . Suppose that the function f is defined for all real numbers . If , then . Also, if , then .
Proof Will be provided later.
NOTE: Let and be sequences with a domain of . Let and where the functions f and g are defined for all real numbers . If the functions f and g are differentiable for all real numbers and if has an indeterminate form of or , then we can apply L’Hopital’s Rule to .
We will need the following theorem in order to determine the limit of certain sequences.
Theorem If is a real number, then
1. if
2. if
3. does not exist if
Proof Will be provided later.
Theorem If , then .
Proof Will be provided later.
Examples Determine whether the following sequences converge or diverge. If the sequence converges, then give its limit.
1.
This sequence is one of our examples given above.
= =
NOTE: If you want to apply L’Hopital’s Rule, then you would have to do the following.
= =
Thus, =
Answer: Converges;
2.
=
We will apply L’Hopital’s Rule to .
= = = = =
Thus,
Answer: Converges; 0
3.
Since and for , then for all .
Since and , then by the Sandwich Theorem.
Answer: Converges; 0
4.
=
We will apply L’Hopital’s Rule to .
= = since since
and
Thus,
Answer: Diverges
5.
NOTE: is undefined when since when . Thus, the domain of is the set .
Now consider .
Since and for , then for all .
Since and = , then by the Sandwich Theorem.
Answer: Converges; 0
6.
=
Since = , then we can write = , which has an indeterminate form of .
We will apply L’Hopital’s Rule to . Thus,
= = =
= =
Thus, . Thus,
Answer: Converges; 3
7.
This is a constant sequence since for all .
Answer: Converges;
8.
Since , then by one of the theorems above, .
Thus,
Answer: Converges; 4
9.
Since , then by one of the theorems above, .
Thus,
Answer: Diverges
10.
Since , then by one of the theorems above, = DNE.
Answer: Diverges
11.
Recall:
NOTE: = =
= for all
Thus, for all
Since and , then by the Sandwich Theorem.
Answer: Converges; 0
Copyrighted by James D. Anderson, The University of Toledo