The value of frac{1}{(1 + a)(2 + a)} + frac{1 | vedclass

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Question

(A) The given series is $S = \sum_{n=1}^{\infty} \frac{1}{(n + a)(n + 1 + a)}$.Using partial fractions,the $n^{th}$ term $T_n$ is given by:$T_n = \fra

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$\frac{1}{1 + a}$

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(A) The given series is $S = \sum_{n=1}^{\infty} \frac{1}{(n + a)(n + 1 + a)}$.Using partial fractions,the $n^{th}$ term $T_n$ is given by:$T_n = \frac{1}{(n + a)(n + 1 + a)} = \frac{1}{n + a} - \frac{1}{n + 1 + a}$.Writing out the sum of the first $n$ terms $S_n$:$S_n = (\frac{1}{1 + a} - \frac{1}{2 + a}) + (\frac{1}{2 + a} - \frac{1}{3 + a}) + (\frac{1}{3 + a} - \frac{1}{4 + a}) + \dots + (\frac{1}{n + a} - \frac{1}{n + 1 + a})$.This is a telescoping series where intermediate terms cancel out:$S_n = \frac{1}{1 + a} - \frac{1}{n + 1 + a}$.To find the sum to infinity,we take the limit as $n 	o \infty$:$S_{\infty} = \lim_{n 	o \infty} (\frac{1}{1 + a} - \frac{1}{n + 1 + a}) = \frac{1}{1 + a} - 0 = \frac{1}{1 + a}$.

The value of $\frac{1}{(1 + a)(2 + a)} + \frac{1}{(2 + a)(3 + a)} + \frac{1}{(3 + a)(4 + a)} + \dots + \infty$ is,(where $a$ is a constant)

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