1. $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n+n}$
2. $\displaystyle\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$
3. $1 - \frac{3}{4} + \frac{4}{6} - \frac{5}{8} + \frac{6}{10} - \cdots$
4. $1 + \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots$
5. $1 - \frac{1}{2^2} + \frac{1}{3} - \frac{1}{4^2} + \frac{1}{5} - \cdots$

1. Two series $\sum x_n$ and $\sum y_n$ that both diverge but where $\sum x_n y_n$ converges.
2. A convergent series $\sum x_n$ and a bounded sequence $(y_n)$ such that $\sum x_n y_n$ diverges.
3. Sequences $(x_n)$ and $(y_n)$ where $\sum x_n$ and $\sum(x_n + y_n)$ both converge but $\sum y_n$ diverges.
4. A sequence $(x_n)$ with $0 \leq x_n \leq 1/n$ where $\sum(-1)^n x_n$ diverges.

Prove or give a counterexample:

1. If $\sum a_n$ converges absolutely, then $\sum a_n^2$ also converges absolutely.
2. If $\sum a_n$ converges and $(b_n)$ converges, then $\sum a_n b_n$ converges.
3. If $\sum a_n$ converges conditionally, then $\sum n^2 a_n$ diverges.

1. An open set containing every rational number must be all of $\mathbb{R}$.
2. Every nonempty open set contains a rational number.
3. Every bounded infinite closed set contains a rational number.

1. $\overline{A \cup B}$
2. $A \setminus B$
3. $(A^c \cup B)^c$
4. $(A \cap B) \cup (A^c \cap B)$
5. $\overline{A}^c \cap \overline{A^c}$

1. Prove that $\overline{A \cup B} = \overline{A} \cup \overline{B}$.
2. Does this extend to infinite unions?