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  4. The sum of the series (1/3× 7)+(1/7× 11)+(1/11× 15)+........ to ∞ is

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Q. The sum of the series $\frac{1}{3\times 7}+\frac{1}{7\times 11}+\frac{1}{11\times 15}+........ to \,\infty$ is

Sequences and Series

Solution:

$T_{n} = \frac{1}{\left[3+\left(n-1\right)4\right]\left[7+\left(n-1\right)4\right]} $

$= \frac{1}{\left(4n-1\right)\left(4n+3\right)} $

$= \frac{1}{4}\left[\frac{1}{4n-1} -\frac{1}{4n+3}\right] $

$\therefore T_{1} = \frac{1}{4}\left[\frac{1}{3}-\frac{1}{7}\right], T_{2}=\frac{1}{4} \left[\frac{1}{7}-\frac{1}{11}\right] $

$ T_{3} = \frac{1}{4}\left[\frac{1}{11}-\frac{1}{15}\right] $ and so on.

$ \therefore S_{\infty} = \frac{1}{4} \left[\frac{1}{3}\right] = \frac{1}{12}$

[All other terms will cancel]