The sum of the infinite series frac{1}{2!} + | vedclass

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(C) The $n^{th}$ term of the series is $T_n = \frac{n}{(n+1)!}$.We can rewrite this as $T_n = \frac{n+1-1}{(n+1)!} = \frac{n+1}{(n+1)!} - \frac{1}{(n+

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(C) The $n^{th}$ term of the series is $T_n = \frac{n}{(n+1)!}$.We can rewrite this as $T_n = \frac{n+1-1}{(n+1)!} = \frac{n+1}{(n+1)!} - \frac{1}{(n+1)!} = \frac{1}{n!} - \frac{1}{(n+1)!}$.Summing from $n=1$ to $\infty$:$S = \sum_{n=1}^{\infty} \left( \frac{1}{n!} - \frac{1}{(n+1)!} \right)$.This is a telescoping series:$S = \left( \frac{1}{1!} - \frac{1}{2!} \right) + \left( \frac{1}{2!} - \frac{1}{3!} \right) + \left( \frac{1}{3!} - \frac{1}{4!} \right) + \dots$$S = \frac{1}{1!} = 1$.

The sum of the infinite series $\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \frac{4}{5!} + \dots$ is

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