Limit of a Sequence: Wayne's Inquiry

In summary, the conversation discusses the concept of limit for a sequence, specifically when the limit is infinite. The speakers mention that in such cases, the sequence should not attain the infinite limit at a finite value, as it would make the sequence uninteresting. However, unless there is a specific context, it should not be assumed that the sequence cannot be of that form.
  • #1
wayneckm
68
0
Hello all, indeed this is always a question in my mind.

For a sequence, we can study the limit, let's say [tex] \lim_{n\rightarrow\infty} x_{n} = c[/tex] where [tex] c [/tex] can be [tex] \infty[/tex].

So whenever we talk about this kind of limit, we are generally interested in a sequence which would not attain [tex] c [/tex] at a finite value of [tex] n [/tex]. In other words, the sequence in the form of [tex] x_{n} = c [/tex] where [tex] n \geq N [/tex] for some finite [tex] N [/tex] is of no interest because the limit is trivial?

Thanks.


Wayne
 
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  • #2
If it becomes constant, then it's a pretty boring sequence, but unless you have some reason to believe so you shouldn't assume that the sequence can't be of that form. Do you have a specific context from which this question is coming?
 

FAQ: Limit of a Sequence: Wayne's Inquiry

What is a limit of a sequence?

A limit of a sequence is a value that the terms of the sequence approach as the index increases. It is the value that the terms of the sequence get closer and closer to, but may never actually reach.

How is the limit of a sequence different from the limit of a function?

The limit of a sequence is similar to the limit of a function, but there are some key differences. For a sequence, the index values (such as n or k) approach a specific value, while for a function, the input values (such as x) approach a specific value. Additionally, the limit of a sequence can exist even if the function is not defined at that limit point, while the limit of a function only exists if the function is defined at that limit point.

Can a sequence have more than one limit?

No, a sequence can only have one limit. This is because the limit represents the value that the terms of the sequence get closer and closer to, and a sequence cannot approach multiple values simultaneously.

What is the importance of understanding limits of sequences?

Limits of sequences are important in many areas of mathematics, including calculus, analysis, and differential equations. They are used to define continuity, convergence, and other mathematical concepts. Additionally, understanding limits of sequences can help us better understand the behavior of functions and their graphs.

How can we determine the limit of a sequence?

To determine the limit of a sequence, we can use various techniques, such as the squeeze theorem, the monotone convergence theorem, or the ratio test. We can also use algebraic manipulation or graphing to gain insight into the behavior of the sequence. In some cases, the limit may be obvious, while in others, it may require more advanced mathematical tools.

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