Given the Point set S - Part (b)

[nasi112]

Given the point set [MATH]S:\{i, \frac{i}{2}, \frac{i}{3}, \frac{i}{4}, . . .\}[/MATH]
(b) What are its limit points, if any?

 

[Deleted member 4993]

Given the point set [MATH]S:\{i, \frac{i}{2}, \frac{i}{3}, \frac{i}{4}, . . .\}[/MATH]
(b) What are its limit points, if any?

What is the definition of "limit points" of a sequence?

I have to repeat myself - again!!

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this problem.

 

[nasi112]

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this problem.

I did show my work before and you ignored it. I am afraid you ignore my working again.

What is the definition of "limit points" of a sequence?

A number [MATH]l[/MATH] is said to be a limit point of a sequence [MATH]u[/MATH] if every neighborhood [MATH]N_l[/MATH] of [MATH]l[/MATH] is such that [MATH]u_n ∈ N_l[/MATH], for infinitely many values of [MATH]n ∈ N[/MATH].

 

[HallsofIvy]

In particular, if a set is sequence and that sequence has a limit then that limit is the only "limit point" of the set. So what is the limit of this sequence?

 

[nasi112]

In particular, if a set is sequence and that sequence has a limit then that limit is the only "limit point" of the set. So what is the limit of this sequence?

No idea. I still didn't grasp the definition. What does neighborhood mean?

 

[pka]

No idea. I still didn't grasp the definition. What does neighborhood mean?

Frankly, it seems to me that you may not be prepared to answer this question.
First you posted the question in the Probability/Statistics sub-forum. It as s nothing to do with either.
Then you don't seem to be completely comfortable with complex number theory.
That is, you seem to not understand the difference in limit of a sequence and the limit point of a set.
If I am mistaken, please correct me by telling us about the context in which you have encountered the question.

 

[nasi112]

Frankly, it seems to me that you may not be prepared to answer this question.
First you posted the question in the Probability/Statistics sub-forum. It as s nothing to do with either.
Then you don't seem to be completely comfortable with complex number theory.
That is, you seem to not understand the difference in limit of a sequence and the limit point of a set.
If I am mistaken, please correct me by telling us about the context in which you have encountered the question.

Would you be more comfortable if I post it in advanced algebra or calculus? Lol. It seems that you guys are not prepared for these types of questions. I would consider sending simple questions like find the integral of [MATH]\int \frac{1}{x} \ dx [/MATH]
It is not shame to confess that this type of questions are above your math level. Even my professor sometimes confesses he gets lost with a simple geometric question that a student in 10th grade can solve it easily.

If you cannot answer the question, don't change the subject by sending show your work, post it in the correct forum, which text-book, you're reading, you're not ready....etc. You will be surprised that this question has to do with probability/statistics as well as complex.

The best answer for these types of question is "Do you know the definition of................?" No. Read it. Lol.

 

[pka]

Would you be more comfortable if I post it in advanced algebra or calculus? Lol. It seems that you guys are not prepared for these types of questions. I would consider sending simple questions like find the integral of [MATH]\int \frac{1}{x} \ dx [/MATH]It is not shame to confess that this type of questions are above your math level. Even my professor sometimes confesses he gets lost with a simple geometric question that a student in 10th grade can solve it easily.
If you cannot answer the question, don't change the subject by sending show your work, post it in the correct forum, which text-book, you're reading, you're not ready....etc. You will be surprised that this question has to do with probability/statistics as well as complex.
The best answer for these types of question is "Do you know the definition of................?"

You did not answer my questions. Please do.
What is the context in which this question occurs?
Do you have an assigned textbook? If so please give its title and author.

 

[nasi112]

You did not answer my questions. Please do.
What is the context in which this question occurs?
Do you have an assigned textbook? If so please give its title and author.

Have you ever seen a Dodo?

 

[HallsofIvy]

No idea. I still didn't grasp the definition. What does neighborhood mean?

So you are saying you don't know the definitions of the words here? Then where did you get this problem? It appears to be homework for a course you aren't taking!

 

[HallsofIvy]

Have you ever seen a Dodo?

I'm beginning to suspect we are dealing with one!

I almost decided not to post this but since I had already written it-

No idea. I still didn't grasp the definition. What does neighborhood mean?

So you are saying you don't know the definitions of the words here? Then where did you get this problem? It appears to be homework for a course you aren't taking!

A "point" is a "limit point" of a set if any neighborhood of that point contains at least one point of the set (and it can be shown then that any neighborhood contains infinitely many points of the set)

A "neighborhood" of a point is the set of all points sufficiently close to that point. More precisely a neighborhood of point "p is the set of all points, x, such that \(\displaystyle |x- p|< \delta\) where [/tex]\delta[/tex] is some small number.
set
If you had taken, or were taking, a course in which "limit" and "limit point" were defined you would have immediately realized that, for any fixed number, i, \(\displaystyle \frac{i}{2}. \frac{i}{3}. \frac{i}{4}, \cdot\cdot\cdot, \frac{i}{n}, \cdot\cdot\cdot\) is a sequence where the numerator is the fixed number, i, while the denominator gets larger and larger- the limit of the sequence is 0 so the only limit point of this set is 0.

Here's a little harder example: Find the limit points (plural!) of the set of numbers 1/2, 3/2, 1/4, 5/4, 1/8, 9/8, 1/16, 17/16, ..., 1/2^n, (2^n+1)/2^n .... With experience (and I have plenty) you would recognize that set as really being two sequences, 1/2, 1/4, 1/8, ..., 1/2^n, ... and 3/2, 5/4, 9/8, 17/16, ..., (2^n+ 1)/2^n.

Now look at those two sequences separately. As in the previous problem, 1/2, 1/4, 1/8, ..., 1/2^n has the numerator fixed, 1, while the denominator gets larger and larger- it also has limit 0. The other sequence, 3/2, 5/4, 9/8, ..., (2^n+ 1)/2^n= 1+ 1/2^n, has 1 plus something that, once again, goes to 0. The two sequences have limits 0 and 1 so the original set has two limit points, 0 and 1.

 

[Deleted member 4993]

I'm beginning to suspect we are dealing with one!

I almost decided not to post this but since I had already written it-


So you are saying you don't know the definitions of the words here? Then where did you get this problem? It appears to be homework for a course you aren't taking!

A "point" is a "limit point" of a set if any neighborhood of that point contains at least one point of the set (and it can be shown then that any neighborhood contains infinitely many points of the set)

A "neighborhood" of a point is the set of all points sufficiently close to that point. More precisely a neighborhood of point "p is the set of all points, x, such that \(\displaystyle |x- p|< \delta\) where [/tex]\delta[/tex] is some small number.
set
If you had taken, or were taking, a course in which "limit" and "limit point" were defined you would have immediately realized that, for any fixed number, i, \(\displaystyle \frac{i}{2}. \frac{i}{3}. \frac{i}{4}, \cdot\cdot\cdot, \frac{i}{n}, \cdot\cdot\cdot\) is a sequence where the numerator is the fixed number, i, while the denominator gets larger and larger- the limit of the sequence is 0 so the only limit point of this set is 0.

Here's a little harder example: Find the limit points (plural!) of the set of numbers 1/2, 3/2, 1/4, 5/4, 1/8, 9/8, 1/16, 17/16, ..., 1/2^n, (2^n+1)/2^n .... With experience (and I have plenty) you would recognize that set as really being two sequences, 1/2, 1/4, 1/8, ..., 1/2^n, ... and 3/2, 5/4, 9/8, 17/16, ..., (2^n+ 1)/2^n.

Now look at those two sequences separately. As in the previous problem, 1/2, 1/4, 1/8, ..., 1/2^n has the numerator fixed, 1, while the denominator gets larger and larger- it also has limit 0. The other sequence, 3/2, 5/4, 9/8, ..., (2^n+ 1)/2^n= 1+ 1/2^n, has 1 plus something that, once again, goes to 0. The two sequences have limits 0 and 1 so the original set has two limit points, 0 and 1.

Prof. Ivey,

I propose we stop wasting our breath (or electrons) - particularly on this thread till and until we are shown a minimum modicum of proficiency.

 

[nasi112]

I'm beginning to suspect we are dealing with one!

I almost decided not to post this but since I had already written it-


So you are saying you don't know the definitions of the words here? Then where did you get this problem? It appears to be homework for a course you aren't taking!

A "point" is a "limit point" of a set if any neighborhood of that point contains at least one point of the set (and it can be shown then that any neighborhood contains infinitely many points of the set)

A "neighborhood" of a point is the set of all points sufficiently close to that point. More precisely a neighborhood of point "p is the set of all points, x, such that \(\displaystyle |x- p|< \delta\) where [/tex]\delta[/tex] is some small number.
set
If you had taken, or were taking, a course in which "limit" and "limit point" were defined you would have immediately realized that, for any fixed number, i, \(\displaystyle \frac{i}{2}. \frac{i}{3}. \frac{i}{4}, \cdot\cdot\cdot, \frac{i}{n}, \cdot\cdot\cdot\) is a sequence where the numerator is the fixed number, i, while the denominator gets larger and larger- the limit of the sequence is 0 so the only limit point of this set is 0.

Here's a little harder example: Find the limit points (plural!) of the set of numbers 1/2, 3/2, 1/4, 5/4, 1/8, 9/8, 1/16, 17/16, ..., 1/2^n, (2^n+1)/2^n .... With experience (and I have plenty) you would recognize that set as really being two sequences, 1/2, 1/4, 1/8, ..., 1/2^n, ... and 3/2, 5/4, 9/8, 17/16, ..., (2^n+ 1)/2^n.

Now look at those two sequences separately. As in the previous problem, 1/2, 1/4, 1/8, ..., 1/2^n has the numerator fixed, 1, while the denominator gets larger and larger- it also has limit 0. The other sequence, 3/2, 5/4, 9/8, ..., (2^n+ 1)/2^n= 1+ 1/2^n, has 1 plus something that, once again, goes to 0. The two sequences have limits 0 and 1 so the original set has two limit points, 0 and 1.

Thank you. This is the same approach I was thinking to follow, but when a fancy tells ( you seem to not understand the difference in limit of a sequence and the limit point of a set ), he lets you think you're doing it wrong.