frac{1}{2!} + frac{1 + 2}{3!} + frac{1 + 2 + | vedclass

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(C) The general term of the series is $T_n = \frac{1 + 2 + \dots + n}{(n + 1)!}$.Using the sum of first $n$ natural numbers,$T_n = \frac{n(n + 1)}{2(n

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$e/2$

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(C) The general term of the series is $T_n = \frac{1 + 2 + \dots + n}{(n + 1)!}$.Using the sum of first $n$ natural numbers,$T_n = \frac{n(n + 1)}{2(n + 1)!} = \frac{n}{2(n + 1)!}$.To simplify,we write $T_n = \frac{(n + 1) - 1}{2(n + 1)!} = \frac{1}{2} \left( \frac{1}{n!} - \frac{1}{(n + 1)!} \right)$.Summing from $n = 1$ to $\infty$,$S = \sum_{n=1}^{\infty} \frac{1}{2} \left( \frac{1}{n!} - \frac{1}{(n + 1)!} \right) = \frac{1}{2} \left[ (\frac{1}{1!} - \frac{1}{2!}) + (\frac{1}{2!} - \frac{1}{3!}) + \dots \right]$.This is a telescoping series,so $S = \frac{1}{2} (1) = \frac{1}{2}$.

$\frac{1}{2!} + \frac{1 + 2}{3!} + \frac{1 + 2 + 3}{4!} + \dots \infty = $

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