Let's analyze the sequence: \[ \{r^n\}_{n=0}^{\infty} \to 0, 1 \] We are to **determine whether the sequence converges or diverges**. If it **converges**, we must **find the limit**. --- ### Step 1: Understanding the Sequence The sequence given is: \[ r^n \text{ for } n = 0, 1, 2, \ldots \] We are told that \( r \) can take values in \( \{0, 1\} \). --- ### Step 2: Analyze for \( r = 0 \) \[ r^n = 0^n \] - For \( n = 0, 0^0 \) is **indeterminate**, but for sequences we usually define \( 0^0 = 1 \). - For \( n > 0 \), \( 0^n = 0 \). So, the sequence is: \[ 1, 0, 0, 0, 0, \ldots \] As \( n \to \infty \), the sequence approaches \( 0 \). #### **Conclusion for \( r = 0 \):** - **The sequence converges to \( 0 \).** --- ### Step 3: Analyze for \( r = 1 \) \[ r^n = 1^n = 1 \text{ for all } n \] So, the sequence is: \[ 1, 1, 1, 1, 1, \ldots \] As \( n \to \infty \), the sequence is constantly \( 1 \). #### **Conclusion for \( r = 1 \):** - **The sequence converges to \( 1 \).** --- ### Step 4: Final Answer #### **Summary Table:** | \( r \) | Sequence | Limit | Converges/Diverges | |---------|----------------------------|-----------|--------------------| | 0 | 1, 0, 0, 0, ... | 0 | Converges | | 1 | 1, 1, 1, 1, ... | 1 | Converges | --- ### **Final Answer** - The sequence \( \{r^n\}_{n=0}^{\infty} \) **converges** for both \( r = 0 \) and \( r = 1 \). - If \( r = 0 \), \( \lim_{n \to \infty} r^n = 0 \). - If \( r = 1 \), \( \lim_{n \to \infty} r^n = 1 \).