Theodore K's question at Yahoo Answers (Radius of convergence)

[Fernando Revilla]

Here is the question:

How do i begin to solve for the Radius of convergence and Interval of this series? I know I should be using either the ratio test or root test for this problem :

(-1)^n * (x^(3n)) / ((2n)!) from n=0 to inf

Here is a link to the question:

Calculus Power Series/Radius of Convergence/Interval of Convergence Question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Theodore K,

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 $R=+\infty$.