# Rates of convergence

## Rates of convergence

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 quadratic 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 r_k =\begin{cases} \frac{1}{k}, & \text{if } k\text{ is even} \\ \frac{1}{k^2}, & \text{if } k\text{ is odd} \end{cases}.
8. Determine the convergence or divergence of a given sequence r_k =\begin{cases} \frac{1}{k^k}, & \text{if } k\text{ is even} \\ \frac{1}{k^{2k}}, & \text{if } k\text{ is odd} \end{cases}.
9. Show that the sequence x_k = 1 + (0.5)^{2^k} is quadratically converged to 1.
10. Determine the convergence or divergence of a given sequence r_k =\begin{cases} \left(\frac{1}{4}\right)^{2^k}, & \text{if } k\text{ is even} \\ \frac{x_{k-1}}{k}, & \text{if } k\text{ is odd} \end{cases}.
11. Let \left\{ r_k \right\}_{k=m}^\infty be a sequence of non-negative numbers and let s > 0 be some integer. Prove that sequence \left\{ r_k \right\}_{k=m+s}^\infty is linearly convergent with constant q if and only if the sequence \left\{ r_k \right\}_{k=m}^\infty converged linearly with constant q.
12. Determine the convergence type of a given sequence r_k =\begin{cases} \frac{1}{2^k}, & \text{if } k\text{ is odd} \\ \frac{1}{3^{2k}}, & \text{if } k\text{ is even} \end{cases}.