How can I test if a sequence is converging?

In summary, the conversation focused on determining if a sequence is converging, with a specific example given. The suggested methods for checking convergence included looking into Cauchy sequences, using the formal definition of convergence, and recognizing that bounded, monotone sequences converge. It was also suggested to read a chapter on sequences in a book on analysis and to use the triangle inequality to confirm convergence. The definition of convergence was also restated as a sequence eventually entering and remaining within any given interval surrounding the proposed limit.
  • #1
FrogPad
810
0
This is related to my senior design project.

I was wondering what would be a good way to check if a sequence is converging. As an example, imagine I have the following:

for n = 1:10,000
x(n) = f(n,n-1)
end

What I'm trying to say is that the sequence is updated in each iteration, and that the value of each sequence element is a function of present and past inputs. I know that the sequence is converging. Let's say that it is converging to zero. However, the sequence may oscillate like crazy, and cross zero many times, but eventually it will settle.

My question:
How can I check if a sequence is converging to zero? What are some standard tests that I can look into?

As of now, I have some hack code that basically computes the moving average. If the moving average gets to some threshold (i.e. not changing much), then I keep checking to see if it drops below some other threshold.
 
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  • #3
It would be worthwhile to pick up a book on analysis and read the chapter on sequences. You would be interested in the formal definition of convergence--

[tex](a_n) \rightarrow A[/tex] if for any [tex]\epsilon > 0[/tex] there exists a natural number [tex]N[/tex] such that [tex]n\geq N \Rightarrow |a_n - A| < \epsilon[/tex].

Another important thing you should know-- bounded, monotone sequences converge.

Many times you can just straight up use the definition to confirm that a sequence converges. The trick is that many times you will set up a triangle inequality to get the job done. Which I even think is how Cauchy sequences (mentioned in the previous post) can be shown to converge.

Let me rephrase that definition-- a sequence is convergent if for any interval around the proposed limit the sequence eventually enters the interval and never leaves. That means for any interval surrounding the proposed limit, there should be an infinite number of terms in the sequence contained within that interval.
 

What is the definition of convergence of a sequence?

The convergence of a sequence refers to the behavior of a sequence of numbers as its terms approach a certain value or limit. In other words, a sequence is said to converge if its terms eventually become closer and closer to a single value as the sequence progresses.

What is the difference between convergent and divergent sequences?

A convergent sequence is one in which the terms eventually become closer and closer to a single value as the sequence progresses. On the other hand, a divergent sequence is one in which the terms do not approach a single value, but instead either increase without bound or oscillate between different values.

What is the importance of understanding convergence of a sequence?

Understanding convergence of a sequence is important in many areas of mathematics and science, including calculus, analysis, and statistics. It allows us to make predictions and draw conclusions about the behavior of a sequence and its limit, which can have real-world applications in fields such as physics and engineering.

What are some common methods for determining convergence of a sequence?

Some common methods for determining convergence of a sequence include the limit comparison test, the ratio test, and the root test. These tests involve comparing the given sequence to known convergent or divergent sequences, and using their behavior to determine the behavior of the given sequence.

Can a sequence be both convergent and divergent?

No, a sequence cannot be both convergent and divergent. A sequence can either converge to a single limit or diverge, but it cannot exhibit both behaviors simultaneously.

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