# Accumulation Points

• Sep 26th 2009, 09:17 AM
zhupolongjoe
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.
• Sep 26th 2009, 10:01 AM
Plato
Quote:

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 $\left\{ {2^{n + \left( {1/k} \right)} :\left\{ {n,k} \right\} \subseteq \mathbb{Z}^ + } \right\}$ is a subset of the positive real numbers.
Because $\left( {3 + \left( {1/k} \right)} \right) \to 3\; \Rightarrow \;\left( {2^{3 + \left( {1/k} \right)} } \right) \to 8$
Therefore, 8 is an accumulation point.
• Sep 26th 2009, 01:21 PM
zhupolongjoe
Thanks, but I guess I wasn't clear.

It is not 2^(n+1/k), it is (2^n)+(1/k)
• Sep 26th 2009, 01:36 PM
Plato
Quote:

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 $2^n+\frac{1}{k}$.
If you say fix $n=3$ then $\left(2^3+\frac{1}{k}\right)\to 8$.
• Sep 26th 2009, 01:50 PM
zhupolongjoe
Ok, but it says to find all the accumulation points, so would it just be all numbers like (2^1, 2^2, .....)?
• Sep 26th 2009, 01:54 PM
Plato
Quote:

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?
• Sep 26th 2009, 02:19 PM
zhupolongjoe
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."