<s>[INST] Study the convergence of the series I need to study the convergence of the following series:
$$\sum_{n=1}^{\infty}(\frac{1\cdot3\cdot5\cdot\cdots\cdot(2n+1)}{1\cdot4\cdot7\cdot\cdots\cdot(3n+1)}x^n)^2$$ I tried to expand that term and reduce some terms but didn't succeed, I also tried to apply the ratio test but still nothing. Any help is much appreciated. 
 [/INST] The ratio test gives you
$${\left(\frac{1 \cdot 3\cdot 5\cdot ...\cdot (2n+1)\cdot (2n+3)}{1\cdot 4\cdot 7\cdot ...\cdot (3n+1)\cdot (3n+4)}x^{n+1}\right)^2\over \left(\frac{1\cdot 3\cdot 5\cdot ...\cdot (2n+1)}{1\cdot 4\cdot 7\cdot ...\cdot (3n+1)}x^n\right)^2}=\frac{(2n+3)^2}{(3n+4)^2} x^2$$
and here you need to take the limit of the ratio. Note that the final result also depends on the value of $x$.
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