The sum of the series frac{1}{3 times 7} + fr | vedclass

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(D) The given series is $S = \frac{1}{3 	imes 7} + \frac{1}{7 	imes 11} + \frac{1}{11 	imes 15} + \dots \infty$.Each term is of the form $\frac{1}{

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$\frac{1}{12}$

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(D) The given series is $S = \frac{1}{3 	imes 7} + \frac{1}{7 	imes 11} + \frac{1}{11 	imes 15} + \dots \infty$.Each term is of the form $\frac{1}{(4n-1)(4n+3)}$.We can write each term as $\frac{1}{4} \left( \frac{1}{4n-1} - \frac{1}{4n+3} ight)$.Thus,$S = \frac{1}{4} \left[ \left( \frac{1}{3} - \frac{1}{7} ight) + \left( \frac{1}{7} - \frac{1}{11} ight) + \left( \frac{1}{11} - \frac{1}{15} ight) + \dots ight]$.This is a telescoping series where all intermediate terms cancel out.$S = \frac{1}{4} \left( \frac{1}{3} - \lim_{n 	o \infty} \frac{1}{4n+3} ight)$.Since $\lim_{n 	o \infty} \frac{1}{4n+3} = 0$,we have $S = \frac{1}{4} 	imes \frac{1}{3} = \frac{1}{12}$.

The sum of the series $\frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} + \dots$ is

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