1. Show with the definition that the sequence $\left\{ \dfrac{1}{k} \right\}_{k=1}^\infty$ does not have a linear convergence rate (but it converges to zero).
2. Show with the definition that the sequence $\left\{ \dfrac{1}{k^k} \right\}_{k=1}^\infty$ does not have a linear convergence rate (but it converges to zero).
3. Determine the convergence or divergence of a given sequence $r_{k} = 0.707^k$.
4. Determine the convergence or divergence of a given sequence $r_{k} = 0.707^{2^k}$.
5. Determine the convergence or divergence of a given sequence $r_{k} = \frac{1}{k^2}$.
6. Determine the convergence or divergence of a given sequence $r_{k} = \frac{1}{k!}$.
7. Determine the convergence or divergence of a given sequence $% $.
8. Determine the convergence or divergence of a given sequence $% $.