Answer
$$\rho = \infty$$
Work Step by Step
1. Use the ratio test to test for convergence:
$$\lim_{n \longrightarrow \infty} \Bigg| \frac{\frac{x^{2(n+1)}}{(n+1)!}}{\frac{x^{2n}}{n!}} \Bigg | = \lim_{n \longrightarrow \infty} \Bigg| \frac{x^{2}}{n + 1} \Bigg| = 0$$
- Therefore, the series converges absolutely when $0 \lt 1$, which is always true, for every x value.
$$\rho = \infty$$
