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

Question

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?

Answer 1

Hint: See Catalan Number with $x=\dfrac14$.