<s>[INST] Infinite Sum of products What is the infinite sum
$$S = {1 + \frac{1}{3} + \frac{1\cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9}+ ....}?$$
I attempted messing around with the $n$ th term in the series but didnt see any solution.
How should I proceed?
 [/INST] HINT:
$n^{th}$ term is $$\frac{1\cdot3\cdot5\cdots(2n-1)}{3^n\cdot n!}$$
$$=\frac{2^n\cdot n!\cdot1\cdot3\cdot5\cdots(2n-1)}{6^n\cdot n!\cdot n!}$$
$$=\frac{2n!}{6^n\cdot n!\cdot n!}$$
$$=\frac{^{2n}C_n}{6^n}$$
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