Interval of convergence 2

Question

How can I find the interval of convergence of:

$$\sum_{k=1}^{\infty} \dfrac {1\cdot 3\cdot 5\cdot \ldots \cdot (2k-1)}{2\cdot 5 \cdot 8 \cdot \ldots \cdot (3k-1)}x^k$$

Thought it can be solved with using integral but I couldn't solve.

Answer 1

Hint: By the ratio test,

$$\sum_{k\ge1}\frac{1\cdot3\cdot5\cdot\cdots\cdot(2k-1)}{2\cdot5\cdot8\cdot\cdots\cdot(3k-1)}x^k$$

converges if

$$\lim_{k\to\infty}\left|\frac{\frac{1\cdot3\cdot5\cdot\cdots\cdot(2k-1)\cdot(2k+1)}{2\cdot5\cdot8\cdot\cdots\cdot(3k-1)\cdot(3k+2)}x^{k+1}}{\frac{1\cdot3\cdot5\cdot\cdots\cdot(2k-1)}{2\cdot5\cdot8\cdot\cdots\cdot(3k-1)}x^k}\right|<1$$