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
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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

Radius of convergence is infinity And interval of convergence is whole real numbers.

Answered: 84252g i find the interval of…

Math

Calculus

84252g i find the interval of canvergence of tbe parer Senes 18

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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