Question

The value of $\lim _{n \rightarrow \infty} \frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots+\frac{1}{(2 n-1)(2 n+1)}$ is

(A) $1 / 2$

(B) $1 / 3$

(C) $1 / 4$

(D) None of these

Answer 1

Let $S_n=\frac{1}{2}\left[\left(1-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{5}\right)+\ldots+\left(\frac{1}{2 n-1}-\frac{1}{2 n+1}\right)\right]$ $\therefore \quad \lim _{n \rightarrow \infty} S_n=\lim _{n \rightarrow \infty} \frac{1}{2}\left[1-\frac{1}{2 n+1}\right]=\frac{1}{2}$

So, The correct option will be (A).