Find the radius of convergence and interval of convergence in the series

Question

Answer 1

First, Use the Ratio Test to determine absolute convergence

a_{n+1 /} a_n = [(n+1)! lxlⁿ⁺¹ / ((1)(3)(5)...(2n-1)(2n+1))] [(1)(3)(5)(7)...(2n-1) / (n lxlⁿ)

= (n+1)lxl / (n(2n+1))

As n goes to infinity, the above expression goes to zero, which is less than 1. So, by the Ratio Test, the series converges absolutely for all x. Consequently, the given series also converges for all x.

Interval of convergence: (-∞, ∞) and Radius of convergence = ∞