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^{2(k+1)}}{2(k+1)!} \times \dfrac{(2k)!}{x^{2k}} \\=\lim\limits_{k \to \infty} \dfrac{x^2}{(k+2)(k+1)}|x|\\=0$
So, we can conclude that the given series converges absolutely for $l=0 \lt 1$ .
Therefore, the interval of convergence for the series is: $(-\infty, \infty)$ and the radius of convergence is $R=+\infty$.