Limit of a Sequence

The notion of limit of a sequence is very natural. Indeed, consider our scientist who is collecting data everyday. Set  to be the sequence generated by our scientist (  is the data collected after n days). Imagine that after a certain day the numbers are very close to each other. Therefore our scientist will decide that the experiment settled down to a equilibrium state, meaning that no change occured to the data. The danger here is that, though the data collected after that date are closer to each other, you should not, in general, believe that the system settles down. Small changes may be responsible for weird behavior. This is the beginning of the Chaos Theory. But, this is not the subject treated here. We will focus more on the nice experiment where the system settles down to an equilibrium state. To better illustrate this phenomena, let us consider the following example.
Example: Take a calculator, set it to "radian mode" and enter the number 1. Then, hit the function Cosine over and over again. Analyze the output of this experiment.
Answer: Then, we have
.
Next, we have
.
If we proceed with this we get

Clearly, the numbers are getting closer to something that starts as 0.73.
To better appreciate the sequence, we graph the points  on a plane (see the Figure below).



Example: Do the same as the example above with the Sine function.
Answer: We have
.
Next, we have
.
If we proceed with this we get

Clearly, the numbers are getting smaller and smaller (see Figure below). In fact, the numbers do get closer to 0 as close as one wishes!!!


Remark: It is amazing to see how slow this sequence gets to 0. There are mathematical reasons behind this which we will not discuss here. But, keep in mind that many people are interested in them (that is, speed of convergence).
After discussing the above two examples one will wonder if any sequence has the same faith (meaning, it gets closer to a number). Unfortunately, the answer is NO. Let us consider a slightly more complicated example.
Example: As before, take your calculator and enter the number 0.3. Second, program your machine to compute y = f(x)= 4(1-x)x. Then, keep on doing the same as you did in the previous two examples. Finally, analyze the output.
Answer: In this case we have  . Then
.
If we keep on iterating, we get

n	xn
1	0.3
2	0.84
3	0.5376
4	0.994345
5	0.0224922
6	0.0879454
7	0.320844
8	0.871612
9	0.447617
10	0.989024

It is clear from this example that no conclusion can be made. In fact, you are right and even more than that; this sequence is completely chaotic (see the Figure below for the first 50 terms of the sequence). Meaning that, even if we compute the first billion terms nothing nice will happen!!! This is a truly scary situation but we will not deal with it here... so don't panic....

By the way, this sequence is used as a discrete mathematical model for population dynamics (called the discrete logistic model).
Let us summarize what we just noticed on these examples.
Consider a sequence of numbers  . Sometimes the numbers get closer and closer to a number L (we will write  ). And sometimes the numbers do not exhibit such behavior. If it does, we say that the sequence  is convergent and has a limit equal to L. We will write
,
or
.
It may happen that we will say n gets larger to express that  . If a sequence is not convergent, it is called divergent.
Let us discuss the above definition. A sequence  is convergent if there exists a number L such that the numbers  get closer and closer to L as n gets larger. We have to make sure that the claim  is justified. That is,  gets really close to L. We do not want  to get close to L and then when you go to  it is not!!! This will be terrible in some serious calculations. So, when we say that  gets close to L, asn gets large, we mean that regardless of how close you want  to be to L, if you go far enough you will get there.... Meaning, if I want my  to be close to L up to 100 Digits, then I am sure if n is big enough, I will get close to L up to 100 Digits (this will happen if  ). In other words, let us set  to be a very small number (which measures the error like  ), then there exists  such that for every , we have


The integer N tells you how far you have to go to get closer to L up to . Meaning that, N is somehow responsible for the speed of the sequence; how fast it goes to its limit...
Definition: The sequence  converges to the number L, if and only if,
for every  , there exists  such that for every 



Some authors will use , instead. No harm is done, do not worry about it.
Example: Show that
.

Answer: Let  . We know that there exists an integer  such that
.
Let  . Then, we have


Remark: Keep in mind that  measures the error between the numbers  and the limit L, while the integer N measures how fast the sequence gets closer to the limit L.