Accumulation points and finding them

• H2Pendragon
In summary, an accumulation point of a set S is a point in the real numbers such that there are infinitely many points in S that are arbitrarily close to it. This definition may be easier to understand than the one given in the book, which can be confusing. To find accumulation points, we can look for points where we can find infinitely many points in S that are arbitrarily close to it. In the given problem, the accumulation points are -1, 0, and 1.
H2Pendragon
This is the definition of accumulation point that my book gives:

A is an accumulation point of $$S \subset \mathbb{R}, \forall \epsilon > 0, S \bigcap B(A;\epsilon)$$ is infinite.

The book I have gives horrible examples on what accumulation points actually are (contradicting itself two out of the three times), but never actually gives instructions on how to find the points.

This is the question I have to solve:

Find the accumulation points of

$$S = \left\{\frac{2}{n} + (1 - \frac{1}{n})cos(\frac{n\pi}{2}) : n \in\mathbb{N}\right\}$$

Can anyone help to actually explain to me, in english, what an accumulation point is? I tried Wikipedia but it's more of this meaningless jargon.

Hopefully, understanding what it is I'm looking for will show me how to answer this question. If not, I could use help there too.

I'd post some relevant work I've done on this problem, but I really have no idea where to start! Why can't analysis books ever actually explain things as if I might actually not understand their initial rambling?

In english, an accumulation point of S is a point A of R such that you can find points of S (different from A) arbitrarily close to A.

Some enlightening examples are:

(1) If S={x} is made of just one point, then it has no accumulation point.
(2) In R, every point is an accumulation point.
(3) The set {1/n: n in N} has 0 as its unique accumulation point.

Why couldn't this book have just said that? That makes complete sense. My book doesn't like making sense. I don't think this was written to be a textbook at all, because it certainly doesn't read like it's trying to teach!

Ok so for the question I'm looking for points a,b,c, etc. where I can find infinitely many points arbitrarily close to those points.

Judging from the set itself, it looks like we get S = {2, 1/2, 2/3, 5/4, 2/5, -1/2, 2/7, 9/8, 2/9, -7/10, 2/11...}

Skrew it, I graphed the rest

It looks like it will just go back to looking like a typical cosine graph (which I guess is to be expected since 2/n and 1/n go to 0)

So if we set $$n_{k} = 2k \Rightarrow\left\{\frac{1}{k} + (1 - \frac{1}{2k})cos(k\pi) \right\} \rightarrow (-1,1)$$

and $$n_{k+1} = 2k+1 \Rightarrow\left\{\frac{2}{2k+1} + (1 - \frac{1}{2k+1})cos(\frac{\pi(2k+1)}{2}) \right\} \rightarrow {0}$$

Ok I think I broke my brain. Did any of that make sense? If it does, are the accumulation points for the set -1,0,1?

If it didn't make any sense (because I'm supposing it doesn't at this point, I've had a bit of a head cold all weekend), where should I actually be heading?

Last edited:
That's perfect.

From the definition of the book, if x is in S, then x is an accumulation point of S

Office_Shredder said:
From the definition of the book, if x is in S, then x is an accumulation point of S

How come?

H2Pendragon said:
This is the definition of accumulation point that my book gives:

A is an accumulation point of $$S \subset \mathbb{R}, \forall \epsilon > 0, S \bigcap B(A;\epsilon)$$ is infinite.

For instance, take S={x}. Then $\forall \epsilon > 0, \ S \cap B(A;\epsilon)=\{x\}$, which is not infinite!

infinite and non-empty look very similar on my laptop screen

1. What are accumulation points?

Accumulation points, also known as limit points, are points in a set where every neighborhood of the point contains infinitely many points of the set. In other words, accumulation points are points that are arbitrarily close to the elements of the set.

2. How do you find accumulation points?

To find accumulation points, you need to first determine the set of points in question and then identify which points are arbitrarily close to the elements of the set. This can be done by analyzing the properties of the set or by using various mathematical techniques such as the Bolzano-Weierstrass theorem or the Cauchy convergence criterion.

3. Can a set have more than one accumulation point?

Yes, a set can have multiple accumulation points. In fact, a set can have infinitely many accumulation points. This depends on the properties and structure of the set in question.

4. What is the difference between an accumulation point and a limit point?

In mathematics, the terms accumulation point and limit point are often used interchangeably. However, some sources define them slightly differently. Some define accumulation points as points that have infinitely many points of the set in every neighborhood, while limit points are points that have at least one point of the set in every neighborhood. In either case, the concept is essentially the same.

5. How are accumulation points related to the concept of convergence?

Accumulation points are closely related to the concept of convergence. In fact, a sequence of points converges to a limit point if and only if the limit point is also an accumulation point of the sequence. This means that accumulation points can be thought of as the "endpoints" of a convergent sequence.

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