Sequences, Index, Member of the sequences, Tend to zero, Tend to infinity, Limits of sequences, Limits and operations.
Let's talk a about sequences. Let's start with what they are good for.
Well, for example, they are good for talking about them.
Here is a sequence.
This is the index of the sequence, and it tells us which member of the sequence we are looking at.
One of the most important properties of a sequence is what happens to it when we look at its members that are farther and farther away.
This sequence, for instance, is approaching zero.
So much so, that we could pick any tiny number, and still, there will be a time when the sequence will get closer than that to zero.
In this case the sequence is said to "converge to" or "tend to" zero, or "the limit of the sequence is zero".
It is denoted by this: or this:
Here comes another sequence.
This sequence converges to zero even more.
Well, these sequences usually converge to zero.
There are then these sequences.
They tend to infinity.
LIMITS OF FAMOUS SEQUENCES
There are then such sequences with radicals.
They also tend to infinity.
And here is the most exciting sequence:
If , then
And now let's see what happens if we add up two sequences.
For example and , then it seems logical that
But unfortunately, life is more complicated.
It is possible, that and .
Where does their sum converge in this case?
Well, the truth is that sequence could converge to negative infinity,
to a specific number,
or to positive infinity.
It is true for sequence as well.
As to their sum, there could be these nine cases.
Let's see them.
If both sequences tend to negative infinity, then their sum does, too.
If one of them tends to A, and the other to negative infinity, then their sum tends to negative infinity.
If one of the sequences tends to negative infinity, and the other to positive infinity, then we simply don't know where their sum tends.
It could be negative infinity
it could be 42
and it could be positive infinity as well.
Filling the rest of the table does not provide many surprises. The lower left corner is also a question mark.
Now let's see what it looks like for the product of two sequences.
Unfortunately there will be many cases here.
Well, this is again something that we simply don't know.
The rest is not too exciting:
And then finally some straightforward cases:
Now we will tackle the worst one: division.
There will be quite a few question marks here.
The first one, right away:
But there are more.
Well, the point of these tables is to help us navigate among the various types of limits.
The question marks aren't too helpful, so we will spend some time on figuring out what can be done in those cases.
We will start with one of the most exciting cases: .
If k IS A SPECIFIC NUMBER
Let's see what should be done with limits of this type:
Here is a case like that: