<s>[INST] Is this series convergent or divergent? Kindly asking, what can I do about series
$$
\left(\frac{1}{3}\right)^2+\left(\frac{1\times 4}{3\times 6}\right)^2+\left(\frac{1\times 4\times 7}{3\times 6\times 9}\right)^2+...+\left(\frac{1\times 4\times 7\times...\times (3n-2)}{3\times 6\times 9\times...\times3n}\right)^2+...$$
Indeed, the ratio test fails. Thank you.
 [/INST] Let $a_n$ be the $n$th term. 
The ratio of successive terms is 
$$\frac{a_{n+1}}{a_n} = \frac{(3n+1)^2}{9(n+1)^2} = 1 - \frac{4}{3n} + O\left(\frac{1}{n^2}\right).$$
Thus, the series converges by Raabe's test, which tells us that if 
$$\left|\frac{a_{n+1}}{a_{n}}\right| \sim 1 - \frac{s}{n} \hspace{5ex}(n\to\infty),$$
then the series converges absolutely if $s>1$, diverges if $s<1$, and may converge or diverge if $s=1$.  
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