# Thread: Accumulation Points

1. ## Accumulation Points

Find the accumulation points of the set {2^n+1/k: n and k are positive integers}

The book gives the definition and then the author going on about how beautiful good proof is....without giving any examples of how to find them.

SO is the accumulation point going to be like a pair (n,k) or how should the answer look?

I know that if it were just for the set {1/k}, we would have the accumulation point be 0. The 2^n grows without limit though and so I don't know how that comes into play.

2. Originally Posted by zhupolongjoe Find the accumulation points of the set {2^n+1/k: n and k are positive integers} SO is the accumulation point going to be like a pair (n,k) or how should the answer look?
This set $\displaystyle \left\{ {2^{n + \left( {1/k} \right)} :\left\{ {n,k} \right\} \subseteq \mathbb{Z}^ + } \right\}$ is a subset of the positive real numbers.
Because $\displaystyle \left( {3 + \left( {1/k} \right)} \right) \to 3\; \Rightarrow \;\left( {2^{3 + \left( {1/k} \right)} } \right) \to 8$
Therefore, 8 is an accumulation point.

3. Thanks, but I guess I wasn't clear.

It is not 2^(n+1/k), it is (2^n)+(1/k)

4. Originally Posted by zhupolongjoe Thanks, but I guess I wasn't clear.
It is not 2^(n+1/k), it is (2^n)+(1/k)
Actually that changes nothing.
For each pair of positive integers we have $\displaystyle 2^n+\frac{1}{k}$.
If you say fix $\displaystyle n=3$ then $\displaystyle \left(2^3+\frac{1}{k}\right)\to 8$.

5. Ok, but it says to find all the accumulation points, so would it just be all numbers like (2^1, 2^2, .....)?

6. Originally Posted by zhupolongjoe Ok, but it says to find all the accumulation points, so would it just be all numbers like (2^1, 2^2, .....)?
Well yes, if you have stated the problem as it appears in the textbook.
BTW. What text is it?

7. It is Edward Gaughan, Introduction to Analysis. Verbatim, the problem reads "Determine the accumulation points of the set {2^n + 1/k:n and k are positive integers."

accumulation, points 