Find the domain of $A(x) = 1 + \frac{x^3}{2\times3}+\frac{x^6}{2\times3\times5\times6} + \frac{x^9}{2\times3\times5\times6\times8\times9} +\space ...$

Question

I'm having trouble in finding the domain of the following series:

$$A(x) = 1 + \frac{x^3}{2\times3}+\frac{x^6}{2\times3\times5\times6} + \frac{x^9}{2\times3\times5\times6\times8\times9} +\space ...$$

I don't see how could I write the denominator for expressing the $a_n$ term and later apply a the ratio test. Could you please help me find this series domain?

Answer 1

It seems that $A_n=\sum_{n \geq 1} a_n x^{3n}$ where $a_0=1$ and: $$a_{n+1}=\frac{a_n}{(n+2)(n+3)}$$ so by the ratio test as: $$\frac{x^3}{(n+2)(n+3)} \to 0 \text{ when } n \to \infty$$ for any $x$ the domain is $\Bbb R$.