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Infinite Numerical Sequences
At the beginning we will mention the definition of a function.
A function is a rule that assigns to each element ๐ in a set ๐ฟ exactly one (and
only one 1! ) element ๐ in a set ๐.
Element ๐ is often denoted by ๐ = ๐(๐).
A function ๐ - often called a mapping - is a relation from a set ๐ฟ to set ๐ in
which no two different ordered pairs have the same first coordinates.
Def:
A function ๐ (a mapping ๐) from a set ๐ฟ โ ๐น to set ๐ โ ๐น is a collection of
ordered pairs (๐, ๐) such that:
2 ) for every ๐ โ ๐ฟ there exists one and only one element ๐ โ ๐ such that
3 ) for every ๐ โ ๐ there exists at least element ๐ โ ๐ฟ such that
2 ) if
โ ๐๐๐๐๐ ๐ and
โ ๐๐๐๐๐ ๐ , then ๐ = ๐
We write:
or:
๐ฟ โ is called independent variable
๐ โ is called dependent variable
We usually consider the functions for which the sets ๐ฟ , ๐ are the sets of real
numbers.
The set ๐ฟ is called domain of the function.
The number ๐ = ๐(๐) is called the value of ๐ at ๐ โ it is read: โ๐ of ๐โ
The set ๐ โ is a set of all possible values of ๐(๐).
The set ๐ is called co-domain of the function.
Consider the relations:
We have three different relations in the picture above.
In the picture a) and c) the relations are the functions but in the picture b) the
relation is not a function.
Remark:
If ๐ is a function , we write ๐ = ๐
to say that
Remark:
We will consider the special functions at the beginning, the functions which
map natural number set ๐ต into another set ๐ (๐ โ ๐น), where ๐ โ is a subset
of the real number set.
Numbers ๐
๐
๐
๐
๐
๐
๐
, โฆ are the terms of the sequence.
๐
โ the first term
๐
โ the second term
๐
โ the third term
๐
โ the ๐
๐กโ
term โ the general term of a sequence
Each term ๐
๐
โ will have successor ๐
๐+๐
. We will consider infinite sequences.
Natural numbers ๐, ๐, ๐,... , ๐ ,... are called the indexes of the terms of the
sequence.
The sequence can be denoted by the symbol { ๐
๐
} or shortly ๐
๐
Remark:
Some sequences can be defined by giving a formula for the ๐
๐กโ
term ๐
๐
Examples:
- Let the general term of the sequence be: ๐
๐
The first terms of the sequence are:
๐
๐
๐
๐
๐
๐
Intuitively we can notice that the terms tend to infinity.
- Let the general term of the sequence be: ๐
๐
๐
๐
The first terms of the sequence are:
๐
๐
๐
๐
๐
๐
Intuitively we can notice that the terms tend to zero (but they will never have
zero value).
The sequence with the general term ๐
๐
๐
๐
is called harmonic sequence.
- Let the general term of the sequence be: ๐
๐
๐
๐+๐
First terms of the sequence are:
๐
๐
๐
๐
๐
๐
Intuitively we can notice that the terms tend to one (but they will never have
value one).
- Let consider the sequence with the general term: ๐
๐
๐
First terms of the sequence are:
๐
๐
๐
๐
๐
๐
๐
Intuitively we know that the terms tend to infinity.
- Let: ๐
๐
๐
The sequence has terms:
๐
๐
๐
๐
๐
๐
๐
The terms alternately are positive and negative.
This is so called alternating sequence.
Alternating sequence is a sequence whose terms change sign.
For example a term ๐ ๐
is positive and the term ๐
๐+๐
negative or vice versa.
- Consider the sequence with the general term: ๐
๐
๐
๐
๐
This is also alternating sequence because:
๐
๐
๐
๐
๐
๐
๐
- Let consider the sequence: ๐
๐
= ๐. The sequence has constant term:
๐
๐
๐
๐
๐
๐
This is so called constant sequence.
๐
= ๐ where ๐ โ ๐น, ๐ โ constant number
Monotone Sequences
Example:
a) Consider the sequence with the general term: ๐
๐
The first terms of the sequence are:
We can easily notice that each next term of the sequence is bigger than the
previous one.
b) Consider the sequence with the general term: ๐
๐
๐
๐
The first terms of the sequence are:
We can easily notice that each next term of the sequence is smaller than the
previous one.
Def:
A Sequence ๐
๐
is said to be increasing if and only if each term of the
sequence is bigger than the previous one.
๐
๐+๐
๐
๐โ๐ต
or:
๐
๐+๐
๐
๐โ๐ต
A Sequence ๐
๐
is said to be decreasing if and only if each term of the sequence
is smaller than the previous one.
๐
๐+๐
๐
๐โ๐ต
or:
๐
is decreasing โ
๐+๐
๐
๐โ๐ต
A Sequence ๐
๐
is said to be nondecreasing (increasing or constant) if and
only if each term of the sequence is bigger than the previous one or exactly the
same.
๐
๐+๐
๐
๐โ๐ต
or:
๐
๐+๐
๐
๐โ๐ต
A Sequence ๐
๐
is said to be nonincreasing (decreasing or constant) if and
only if each term of the sequence is smaller than the previous one or exactly
the same.
๐
๐+๐
๐
๐โ๐ต
or:
๐
๐+๐
๐
๐โ๐ต
Example:
Consider the sequence with the general term: ๐
๐
๐
๐
๐
๐
+๐
We will check if the sequence is monotone.
We must check the difference: ๐
๐+๐
๐
The term ๐
๐+๐
is:
๐
๐
๐
๐
๐
+๐
๐+๐
(๐+๐)
๐
๐ (๐+๐)
๐
๐
๐
+๐๐+๐
๐
๐
+๐๐+๐+๐
๐
๐
+๐๐+๐
๐
๐
+๐๐+๐
We count the difference:
๐+๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
Hence: ๐
๐+๐
๐
๐ for all ๐ โ ๐ต.
It means that the sequence ๐ ๐
is increasing.
Exercise:
Determine if the sequence is increasing or decreasing or not monotonic:
๐
๐
๐+๐
Solution:
๐
๐
๐+๐
๐+๐
๐
๐+๐+๐
๐
๐+๐
We check the difference:
๐+๐
๐
๐+๐
๐
< ๐ for all ๐ โ ๐ต, so the sequence ๐
๐
is decreasing
Exercise:
Determine if the sequence is increasing or decreasing or not monotonic:
๐
๐
๐
Solution:
๐
๐
๐
๐+๐
๐
๐+๐
We must count the difference:
๐+๐
๐
๐+๐
๐
< ๐ for all ๐ โ ๐ต so the sequence ๐
๐
is decreasing.
Example:
Let us consider the sequence with the main term: ๐ ๐
๐
We begin by computing the first several terms of the sequence:
A sequence ๐
๐
๐
is bounded below if there is a real number ๐ such that
all terms of the sequence are greater or equal to this number ๐.
The number ๐ = ๐ is lower bound of the sequence.
We can write:
๐
๐
is bounded below โ because
๐=๐
๐โ๐ต
๐
๐
A sequence ๐
๐
is not bounded above because there is no a real number ๐ด
such that all terms of the sequence are less or equal to this number ๐ด.
Example:
Let us consider the sequence with the main term: ๐
๐
๐
๐
โ harmonic
sequence
Some first several terms of the sequence are:
A sequence is bounded below if there is a real number ๐ such that all terms of
the sequence are greater to this number ๐.
The number ๐ = ๐ is lower bound of the sequence.
At the same time, a sequence ๐
๐
is bounded above because there is a real
number ๐ด such that all terms of the sequence are less or equal to this number
The number ๐ด = ๐ is upper bound of the sequence.
So: ๐ = ๐ and ๐ด = ๐.
A sequence ๐
๐
๐
๐
is bounded below and above so it is bounded.
We can write:
๐
๐
๐
is bounded because:
๐=๐
๐ด=๐
๐โ๐ต
๐
or analogously :
๐
๐=๐ ๐ด=๐ ๐๐๐ต
๐=๐๐ด=๐ ๐๐๐ต
Example:
Let us consider the sequence with the main term: ๐
๐
๐
alternating sequence
Some first several terms of the sequence are:
A sequence is bounded below and above because there exists a real number
๐ = โ๐ such that all terms of the sequence are greater or equal to this
number ๐ and there exists a real number ๐ด = ๐ such that all terms of the
sequence are smaller or equal to this number.
The number ๐ = โ ๐ is lower bound of the sequence.
The number ๐ด = ๐ is upper bound of the sequence.
Hence:
๐
๐=โ๐๐ด=๐ ๐๐๐ต
Sequence is bounded.
๐
๐
๐
๐
๐
๐
๐
๐
๐
๐
We have simple estimation:
๐
๐=๐๐ด=๐ ๐๐๐ต
The sequence is bounded.
The number ๐ = ๐ is lower bound of the sequence.
The number ๐ด = ๐ is upper bound of the sequence.
The limit of the sequence
The limit of the sequence is one of the most important notion in mathematics.
The basic question about a sequence ๐
๐
concerns the behavior of its ๐
๐๐
๐๐๐๐ if ๐ gets larger and larger.
Def: Cauchy definition of the limit
The sequence ๐
๐
converges to a real number ๐ , ๐ โ ๐น , if and only if for
each real number ๐บ > ๐ , there exists a positive integer number ๐น , such that
for all numbers ๐ > ๐น , the following inequality is held:
๐
We can write the above definition shortly:
๐โโ
๐
๐บ > ๐
๐น
๐>๐น
๐
We can apply the following notation:
๐โโ
๐
๐
๐
๐ โ โ
Remark:
We say that the sequence ๐
๐
converges to the real number ๐, or tends to the
limit a and we write ๐๐๐
๐โโ
๐
= ๐ if for every ๐บ > ๐ , there is an
integer number ๐น such that
๐
< ๐บ whenever ๐ > ๐น.
Remark:
The points on the graph above of the ๐
๐
must lie between the horizontal lines
๐ = ๐ โ ๐บ and ๐ = ๐ + ๐บ if ๐ > ๐น.
The picture must be valid no matter how small ๐บ is choosen, but usually a
smaller ๐บ requires a larger ๐น.
The sequence ๐ ๐
has a limit equal to ๐ if almost all terms of the sequence
are in the neighborhood of ๐ with radius epsilon ๐บ.
/That is, a sequence ๐
๐
tends to ๐ if and only if every neighborhood of ๐
contains all but a finite number of terms of the sequence/.
In other words:
Sequence ๐
๐
has a limit ๐ if the terms with sufficiently big indexes ๐ differ
from the number ๐ very little.
That is, a sequence {an} converges to a if and only if every neighborhood of a
contains all but a finite number of terms of the sequence
Def:
We say that the sequence ๐
๐
converges if it has a limit which is a real number.
We say the sequence ๐
๐
is convergent โ it has a limit ๐ โ ๐น
We say that the sequence ๐ ๐
diverges if it does not converge to any real
number.
We say in this case that the sequence ๐
๐
is divergent โ a limit ๐ does
not exist or is equal to +โ or โโ.
Example:
We will show using Cauchy definition that the sequence with the main term
๐
๐
๐
has a limit zero , it means that:
๐โโ
Using Cauchy definition, we have :
๐โโ
๐
๐
๐บ > ๐ ๐น ๐>๐น
We can write that:
๐โโ
๐บ > ๐ ๐น ๐>๐น
We will solve the inequality:
Since the solution of the inequality is:
๐
๐บ
, hence we can assume that ๐น =
๐
๐บ
Simultaneously we showed that:
๐>๐น=
๐
๐บ
๐น=
๐
๐บ
๐บ > ๐
It means we proved that:
๐โโ
