Let's carefully read the problem and provide a step-by-step solution. --- **Problem Restatement:** Suppose the series \(\sum_{n=0}^\infty a_n x^n\) **converges for \(|x|<13\)**. Which of the following statements are true? (Select all that apply). --- ### **Step 1: Interpret the Condition** The series \(\sum_{n=0}^\infty a_n x^n\) converges for \(|x|<13\). This means the **radius of convergence \(R\) is 13**. ### **Step 2: Analyze Each Statement** #### **Option 1**: \(\sum_{n=0}^\infty a_n (1.3)^n\) converges. - Here, \(x = 1.3\), and \(|1.3| < 13\). - **True**, because it is within the radius of convergence. #### **Option 2**: \(\sum_{n=0}^\infty a_n (15)^n\) converges. - Here, \(x = 15\), and \(|15| > 13\). - **False**, since it is outside the radius of convergence. #### **Option 3**: \(\sum_{n=0}^\infty a_n (15)^n\) diverges. - Here, \(x = 15\), and \(|15| > 13\). - **True**, since the series diverges outside the radius of convergence. #### **Option 4**: \(\sum_{n=0}^\infty a_n (1.3)^n\) diverges. - Here, \(x = 1.3\), and \(|1.3| < 13\). - **False**, since the series converges for these \(x\). #### **Option 5**: The radius of convergence of the series \(\sum_{n=0}^\infty a_n x^n\) is 26. - **False**, the radius is given as 13. --- ### **Step 3: Final Answer** **The true statements are:** - \(\sum_{n=0}^\infty a_n (1.3)^n\) converges. - \(\sum_{n=0}^\infty a_n (15)^n\) diverges. --- ## **Final Answer** - **Option 1** (converges for \(x=1.3\)) - **Option 3** (diverges for \(x=15\))