Let’s first try to notice some important characteristics of sequences. For sequence:
we can see that members of it are constantly growing, and for sequence
we can see that the members are constantly reducing.
Of course, there are many sequences where we can’t find any of that regularities such as alternating sequences.
Sequence of real numbers will monotonic increase if for every .
Sequence of real numbers will monotonic decrease if for every .
The sequence is strictly monotonic if sign and is sign or .
Notice that for a proof that a sequence is monotonic increasing it is enough to show that
If , then the sequence is decreasing, and if changes its sign for some values of , then the sequence is not monotonic sequence.
Example 1. Is sequence whose general member is monotonic?
Let’s find the difference
Since , or which means that the sequence is monotonic increasing.
Let’s notice some more properties. For example, what do you notice for a sequence given with general member
Probably, just from a look of it, nothing much. Let’s write some of its first members.
We can notice that the members of this sequence never go over , but never decrease below . This means that for every member of this sequence is valid that:
Here we came to second important sequence characteristic:
Sequence is bounded if there exist numbers and such that for every it is true that:
Number is the lower bound and is the upper bound of the set .
We say that the sequence is bounded if we can find its lower and upper bound. Every member of bounded sequence are found in segment .
If we know that a sequence is bounded, we can find infinitely many upper and lower bounds.
Example 2. For the following sequences determin bound?
Let’s write again first few members.
We can clearly see that one of the lower bounds is zero and one of the upper is .
Don’t let this minus frighten you. Just write first few members and conclude what will happen to this sequence.
From here we can see that this sequence is bounded by the first two members. If we’d continue writing these members they would be approaching zero.
This means that .
Again, we’ll write down first few members.
This sequence has no bounds.
One of the most important questions we can ask when observing infinite sequence is:
What happens to the members of a sequence for great values of ?
Let’s observe sequence that is given with his general member .
We can see that the larger the is, the value of sequences member is closer to .
We say that number 1 is the limit value or limit of a sequence . This statement is noted as:
To be accurate this is read:
The limit value of sequence when tends to infinity is equal to .
What about alternating sequences? They also can have limits. Let’s observe sequence
, , , , , , …
This sequence is bounded, for every n. When n grows members of this sequences are approaching zero and the distance of nth member from zero gets smaller and smaller.
This means that
For a sequence of real numbers is said to be convergent if there exists real number such that sequence tends to this number when grows infinitely.
We say that is the limit of a sequence and write .
Can we define limit value in some other way?
If we know that that a limit is a number to which the sequence is tending we can rephrase that sentence in some other way. If is a limit of a sequence, then as grows, the distance between the member of that sequence and number is decreasing.
This means that we can find as small number as we’d like, such that the distance from the members of the sequence to number is smaller than for almost every (all but finitely many) member of the sequence or:
Geometrically watched, condition means that there is as small interval as we’d like that contains every member but finitely many.
Real number is the limit of a sequence of real numbers if for every exists natural number such that for every it is valid that:
For a sequence that is not convergent we say that it is divergent.
Let’s check that the sequence , has limit that is equal to using this second definition. We can imagine a small number , for example .
From here, we can conclude that for every the distance of number to the member of the sequence will be lesser than ; every member of a sequence (except for the first 1000) is in environment of number .