<s>[INST] Evaluate the Sum $S=\frac{1}{4}+\frac{1.3}{4.6}+\frac{1.3.5}{4.6.8}+\cdots \infty$ Evaluate the Sum 

$$S=\frac{1}{4}+\frac{1.3}{4.6}+\frac{1.3.5}{4.6.8}+\cdots \infty$$

My try: We have the $n$ th term as
$$T_n=\frac{1.3.5. \cdots (2n-1)}{4.6.8 \cdots (2n+2)}$$ $\implies$
$$T_n=\frac{1.3.5. \cdots (2n-1)}{2^n \times (n+1)!}$$
$$T_n=\frac{(2n)!}{4^n \times n! \times (n+1)!}$$
$$T_n=\frac{\binom{2n}{n-1}}{n \times 4^n}$$
Any clue here?
 [/INST] Hint: See Catalan Number with $x=\dfrac14$.
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