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Let \( S_{n}=\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\frac{1}{5 \times 7}+ \).to \( n \) terms, then \( \lim _{n \rightarrow \infty} S_{n} \) is equal to |
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Answer» \(S_n = \frac1{1\times 3} + \frac1{3\times 5} + \frac1{5\times 7} + .... + \frac1{(2n - 1 )\times (2n + 1)}\) \( = \frac12\left( \frac2{1\times 3} + \frac2{3\times 5} + \frac2{5\times 7} + .... + \frac2{(2n - 1 )\times (2n + 1)}\right)\) \( = \frac12\left( \frac{3-1}{1\times 3} + \frac{5-3}{3\times 5} + \frac{7-5}{5\times 7} + .... + \frac{(2n + 1)- (2n - 1)}{(2n - 1 )\times (2n + 1)}\right)\) \(= \frac12 \left(\left(1 - \frac13\right) + \left(\frac13 - \frac15\right) + \left(\frac15 - \frac17\right) + ....+ \left(\frac1{2n - 1} - \frac1{2n + 1}\right)\right)\) \(= \frac12 \left(1 - \frac1{2n + 1}\right)\) \(\lim\limits_{n \to \infty} S_n = \lim\limits_{n \to \infty }\frac12 \left( 1 - \frac1{2n + 1}\right) \) \(= \frac12 \left(1 - \frac1\infty\right)\) \(= \frac12 (1 - 0)\) \(= \frac12\) |
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