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The ratio test works well here $$\lim_{n\to +\infty}\left|\frac{u_{n+1}}{u_n}\right|=\lim_{n \to +\infty}\left|\frac{(-1)^{n+1}x^{3n+3}}{(2n+2)!}\cdot\frac{(2n)!}{(-1)^nx^{3n}}\right|=\\\lim_{n\to +\infty}\left|\frac{x^{3}}{(2n+2)(2n+1)}\right|=0<1\; (\forall x\in\mathbb{R})$$ This implies that the radius of convergence is $R=+\infty$.