Download NUMERIC SEQUENCES

NUMERIC SEQUENCES
To analyze behavior of functions nearby a point, we need to evaluate the function for
values closer and closer to the point. It is normally obtained by producing lists of
numbers. Many times we are going to be working with lists of numbers to analyze
mathematical concepts.
These are some listing of numbers. They can be finite or infinite
 Finite
o {1, 3, 5, 7}
 First term
1 denoted
a1 =1
 Second term 3 denoted
a2 = 3
 Third term
5 denoted
a3 = 5
 Fourth term 7 denoted
a4 = 7
 Notice that the terms have the form 2n+1, for n = 0, 1,.., 3. In this
case we can represent the sequence in two forms
an 4n1 or2n  14n1
Observe that {1, 3, 5, 7} refers to the general listing. The notation a1, a2, a3, a4 indicates
that there are four terms, and the sub index specifically indicates the position of the term.
EXERCISE:
o {-2, 0, 2, 4, 6}
 Identify the general term.
o {9, 12, 15}
 Identify the form for the general term
o {-1, -1, -1, -1, -1, -1 } Constant finite sequence
 b1= -1, b2 = -1, …, b6 = -1
 General term is -1, for any value of k = 1, 2, .., 6 or
6
6
bk k 1 or 1k1
 1 1 1
1 
o 1, , , , , 
 2 3 4
20 
 c1=1, c2=1/2, …, c9=
last term is _____
 Form of the general term ___________
 Notation in braces indicating the general term ________________
o {1, -2, 3, -4, 5, -6}
 Form for the general term ___________________
 Notation in braces
1 2 3
30 
o  , , , , 
2 3 4
31 
 General term
 Notation in braces
Sequences 1
1
o Sometimes is hard to find a pattern. For example {1, 3, 2, 9, 1, 1}
In this case we just list the term as a1 1,a2  3,a3  2,...,etc

Infinite Sequences.
o Counting numbers {1, 2, 3, 4, 5, 6…}
 a1=1, a2 = 2, a3 = 3 …
 The 10th term is _________
 General term an  n , n=1, 2, 3, …


 Notation an n1 or nn1
o Consider the sequence 10,11,12,13,...
 Find the general term
 Write in braces notation
Hilbert Hotel paradox:
There is a mountain hotel in Alps, which has infinite number of rooms but all
of them are occupied. A tired group of 3 hikers came to the owner asking to
stay over. He smiled and said, it’s simple, in no time there will be 3 free rooms.
He asked the person from the first room to move to the 4th one, the person from
the 2nd to the 5th and the one from the 3rd to the 6th one. Naturally, the occupant
from the 4th one had to move to the 7th one, and so on. In no time there were 3
rooms free ready for the hikers. How come this could happen?
o Other infinite sequences
 1 1 1 
o 1, , , , 
 2 `3 4 
 Form of general term
 Braces notation
2 3 4 
o  , , ,....
3 4 5 
 Form for the general term
 Braces notation
o 0.5,0.5,0.5,......
 Form for the general term
 Braces notation
 1 1 1 
o 1, , , ,....
 2 4 8 
 General term
 Braces notation
Sequences 1
2
Infinite sequences as functions with discrete domain
.As we learned before, any listing of numbers can be represented as a function with
discrete domain. For example the sequence defined by an  n 1, n 1 , can be thought as
the function f n  n  1 where the domain is the natural numbers. This would be a graph
of this sequence as a function. What happens as n ? Can you explain that?
Graph the following sequences on the number line and as functions. Observe what
happens to them as n 
1
o an  , n 1
n
5
o an 
, n 1
n7
n 2n 1
o an  1
,n 1
n2
For each of those sequences find the terms a100, an 1, a2k
Other examples.
o A calculator gives you the following number when asked for the value
of 2 : 2 = 1.4121356237309…
What’s the meaning of that number?
This statement can also be understood as the infinite sequence of better and better
decimal approximations to the number 2 :
1, 1.4, 1.41, 1.412, 1.4121…
1
 0.333333... can be seen as the sequence of better and better
3
1
decimal approximations 0.3, 0.33, 0.333, 0.3333, … to the number
3
o Similarly
HOMEWORK:
Sequences 1
3
For each of the sequences
2
o an  n ,n  1
1
o bn  2 , n 1
n 1
n
o an  1 ,n  1
n2  1
,n  1
o cn 
n 1
1n
o dn  n ,n  1
2
4n 10
o an 
,n 1
2n
Find
o The 10th, 150th, 1000th terms
o Generate a table of values to predict the behavior of the sequence
when n .
o Graph the sequences to verify your observations from the table you
created. At this point it is impossible to verify your claims, but at least you
can get the feeling for whether or not it is valid.
o What can you say about the values of the sequence when n gets larger and
larger?
o Look at the values of the sequence and compare those values to the end
behavior of the sequence as a function. These are observations, which are
very important for later on.
The familiar paradox
If a runner is to reach the end of the track, he must first complete an infinite number of
different journeys: getting to the midpoint, then to the point midway between the
midpoint and the end, and so on. Hence, the runner cannot complete the race.
Sequences 1
4