Limit of a Sequence

The notion of limit of a sequence is very natural. Indeed, consider our scientist who is collecting data everyday. Set tex2html_wrap_inline233 to be the sequence generated by our scientist ( tex2html_wrap_inline235 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

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Next, we have

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If we proceed with this we get

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Clearly, the numbers are getting closer to something that starts as 0.73.

To better appreciate the sequence, we graph the points tex2html_wrap_inline247 on a plane (see the Figure below).



Example: Do the same as the example above with the Sine function.

Answer: We have

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Next, we have

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If we proceed with this we get

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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 tex2html_wrap_inline259 . Then

displaymath261.

If we keep on iterating, we get

nxn
10.3
20.84
30.5376
40.994345
50.0224922
60.0879454
70.320844
80.871612
90.447617
100.989024
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 tex2html_wrap_inline233 . Sometimes the numbers get closer and closer to a number L (we will write tex2html_wrap_inline267 ). And sometimes the numbers do not exhibit such behavior. If it does, we say that the sequence tex2html_wrap_inline233 is convergent and has a limit equal to L. We will write

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or

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It may happen that we will say n gets larger to express that tex2html_wrap_inline277 . If a sequence is not convergent, it is called divergent.

Let us discuss the above definition. A sequence tex2html_wrap_inline233 is convergent if there exists a number L such that the numbers tex2html_wrap_inline235 get closer and closer to L as n gets larger. We have to make sure that the claim tex2html_wrap_inline267 is justified. That is, tex2html_wrap_inline235 gets really close to L. We do not want tex2html_wrap_inline289 to get close to L and then when you go to tex2html_wrap_inline291 it is not!!! This will be terrible in some serious calculations. So, when we say that tex2html_wrap_inline235 gets close to L, as n gets large, we mean that regardless of how close you want tex2html_wrap_inline235 to be to L, if you go far enough you will get there.... Meaning, if I want my tex2html_wrap_inline235 to be close to L up to 100 Digits, then I am sure if n is big enough, I will get tex2html_wrap_inline235 close to L up to 100 Digits (this will happen if tex2html_wrap_inline305 ). In other words, let us set tex2html_wrap_inline307 to be a very small number (which measures the error like tex2html_wrap_inline309 ), then there exists tex2html_wrap_inline311 such that for every tex2html_wrap_inline313, we have

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The integer N tells you how far you have to go to get closer to L up to tex2html_wrap_inline319. Meaning that, N is somehow responsible for the speed of the sequence; how fast it goes to its limit...

Definition: The sequence tex2html_wrap_inline233 converges to the number L, if and only if,
for every tex2html_wrap_inline307 , there exists tex2html_wrap_inline311 such that for every tex2html_wrap_inline313

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Some authors will use tex2html_wrap_inline331, instead. No harm is done, do not worry about it.

Example: Show that

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Answer: Let tex2html_wrap_inline307 . We know that there exists an integer tex2html_wrap_inline311 such that

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Let tex2html_wrap_inline313 . Then, we have

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Remark: Keep in mind that tex2html_wrap_inline319 measures the error between the numbers tex2html_wrap_inline235 and the limit L, while the integer N measures how fast the sequence gets closer to the limit L.

For more on Limit of Sequences, click HERE.

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