<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] $$\frac{1\times 4\times 7\times\cdots\times (3n-2)}{3\times 6\times 9\times\cdots\times3n}\leqslant\frac1{(n+1)^{2/3}}$$
</s>