Need help on this question Title: Determining the Interval of Convergence for a Power Series### Problem Statement:Find the interval of convergence of the following power series:\[\sum_{{n=1}}^{\infty} \frac{{n!}(x+3)^n}{(2n)!}\]### Explanation:The image presents a series that requires finding the interval of convergence. This involves determining the set of values for \(x\) where the series converges. Use standard tests for convergence, such as the ratio test or root test, to find the interval. #### Series Breakdown:- The numerator includes \(n! \cdot (x+3)^n\), indicating factorial growth and a variable term.- The denominator \((2n)!\) features a factorial term growing with double the index \(n\).#### Steps to Solution:1. **Apply the Ratio Test:** - Define the general term \(a_n = \frac{{n!}(x+3)^n}{(2n)!}\). - Consider the ratio \(\left|\frac{a_{n+1}}{a_n}\right|\) for the series. 2. **Simplify and Analyze:** - Simplify the ratio \(\left|\frac{{(n+1)!}(x+3)^{n+1}}{(2n+2)!} \cdot \frac{(2n)!}{{n!}(x+3)^n}\right|\). - Analyze the limit of the simplified form as \(n \to \infty\).3. **Determine Convergence:** - Use the limit to find values of \(x\) that ensure the ratio is less than 1.4. **Interval of Convergence:** - Solve inequalities derived from the ratio limit to identify open intervals. - Test endpoints if necessary.By conducting these steps, you'll determine the series' interval of convergence and understand behaviors across different \(x\) values.

Name: Need help on this question Title: Determining the Interval of Converge
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Need help on this question Title: Determining the Interval of Convergence for a Power Series### Problem Statement:Find the interval of convergence of the following power series:\[\sum_{{n=1}}^{\infty} \frac{{n!}(x+3)^n}{(2n)!}\]### Explanation:The im