**Problem Statement:**Find the radius of convergence for the series:\[\sum_{n=1}^{\infty} \frac{(n!)^2 x^n}{(2n)!}\]**Instructions:**- Use appropriate convergence tests to determine the radius of convergence.- Input your final answer in the provided box.--------------------------------------------------------------------**Detailed Explanation:**The series given is:\[\sum_{n=1}^{\infty} \frac{(n!)^2 x^n}{(2n)!}\]To find the radius of convergence, consider using the ratio test or the root test. This requires examining the behavior of the terms as \( n \to \infty \). The ratio test considers:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where \( a_n = \frac{(n!)^2 x^n}{(2n)!} \).Calculate:\[a_{n+1} = \frac{((n+1)!)^2 x^{n+1}}{(2(n+1))!}\]Substitute and simplify the following expression for the ratio test:\[\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{((n+1)!)^2 x^{n+1}}{(2(n+1))!} \cdot \frac{(2n)!}{(n!)^2 x^n} \right|\]Finally, solve the limit as \( n \to \infty \) to determine the radius of convergence.

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Answered: Find the radius of convergence for:…

Find the radius of convergence for: (n!)?r" Σ (2n)! n=1

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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