- Decide if the following sequences converge or diverge. If they converge, can you calculate their limit? \[ \left\{(-1)^n\right\} \qquad \left\{\frac{-1}{2^n}\right\} \qquad \left\{\left(\frac{-1}{2}\right)^n\right\} \qquad \left\{\left(\frac{-1}{2}\right)n\right\} \] \[ \left\{\sin(n)\right\} \qquad \left\{\sin(2n)\right\} \qquad \left\{\sin(2\pi n)\right\} \qquad \left\{\sin(\pi n)\right\} \qquad \left\{\cos(\pi n)\right\} \] \[ \left\{\frac{n!}{n^n}\right\} \qquad \left\{\frac{n^n}{n!}\right\} \qquad \left\{\frac{1}{1+n}\right\} \qquad \left\{1+\frac{1}{n}\right\} \qquad \left\{\left(1+\frac{1}{n}\right)^n\right\} \]
- For each of the following explicit sequences whose first few terms are listed, come up with a “simple” closed-form formula for the \(n\text{th}\) term of the sequence. Assume the first term corresponds to \(n=1.\) \[ \left\{\, \frac{2}{3},\, \frac{4}{9},\, \frac{8}{27},\, \frac{16}{81},\,\dotsc \,\right\} \qquad \left\{\, -\frac{2}{3},\, \frac{4}{9},\, -\frac{8}{27},\, \frac{16}{81},\,\dotsc \,\right\} \] \[ \left\{\, \frac{2}{3},\, -\frac{4}{9},\, \frac{8}{27},\, -\frac{16}{81},\,\dotsc \,\right\} \qquad \left\{\, -1,\, \frac{2}{3},\, -\frac{4}{9},\, \frac{8}{27},\, -\frac{16}{81},\,\dotsc \,\right\} \] \[ \left\{\, \frac{2}{3},\, 0,\, -\frac{8}{27},\, 0,\, \frac{32}{243},\, 0,\,\dotsc \,\right\} \qquad \left\{\, \frac{2}{3},\, 0,\, \frac{8}{27},\, 0,\, \frac{32}{243},\, 0,\,\dotsc \,\right\} \]
- For which values of \(r\) will the sequence \(\{r^n\}_{n=1}^{\infty}\) converge?