Answer
The interval of convergence for the series is: $[-\infty, +\infty]$ and the radius of convergence is $R=+\infty$.
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$
1. For $l \lt 1$, the series is absolutely convergent.
2. For $l \gt 1$, the series is divergent.
3. For $l = 1$, the series is inconclusive.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|\\=\lim\limits_{k \to \infty} \dfrac{x^{2k+3}}{(2k+3)!} \times \dfrac{(2k+1)!}{x^{2k+1}} \\=\lim\limits_{k \to \infty} \dfrac{x^2}{(2k+3)(2k+2)}\\=0$
So, we can conclude that the given series converges absolutely for all $x$ when $l \lt 1$.
Therefore, the interval of convergence for the series is: $[-\infty, +\infty]$ and the radius of convergence is $R=+\infty$.
