Tuesday, September 25, 2018

sequences and series - 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?

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