<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] The $n+1$ th term is $$\frac{1\cdot 3\cdot 5\cdots(2n-1) }{3^{n}(n)!}=\frac{(2n)!}{6^{n}(n!)^2}$$ Hence the series is $$\lim_{x\to 1/6}\sum_{n\ge 0}\binom{2n}{n}x^n=\frac{1}{\sqrt{1-4/6}}=\sqrt{{3}}$$ This type of sum appears while calculating probability of return to origin in $1D$ random walks.
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