Chapter 4.3, Problem 6E

Find the derived set of each of the following sets.

1. $\left\{\frac{n+1}{2n}:n\in \mathbb{N}\right\}$
2. $\left\{2^n:n\in \mathbb{N}\right\}$
3. $\left\{6n:n\in \mathbb{N}\right\}$
4. $\left\{\frac{7}{2^n}:n\in \mathbb{N}\right\}$
5. $\left(0,1\right]$
6. $\left(3,7\right)\cup \left\{2,6,8\right\}$
7. $\left\{1+\frac{{\left(-1\right)}^{n}n}{n+1}:n\in \mathbb{N}\right\}$
8. $\mathbb{Q}\cap \left(0,1\right)$
9. $\left\{\frac{1+n^2\left(1+{\left(-1\right)}^n\right)}{n}:n\in \mathbb{N}\right\}$
10. $\left\{\sin x:x\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)\right\}$
11. $\left\{k+\frac{1}{n}:k,n\in \mathbb{N}\right\}$
12. $\left\{\frac{x}{2^y}:y\in \mathbb{Z}\right\}$