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I have the series

$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \text{... }+ \frac{n}{(n+1)!}$

How can I create a compact expression for the sum of the series? Essentially, what is a compact expression for the following?

$$\sum_{i=1}^{n}{\frac{i}{(i+1)!}}$$

I'm lost as to where to start.

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    $\begingroup$ Did you miss $\frac{2}{3!}$ or was it not part of the series? If it wasn't part of the series, then your summation representation is incorrect. $\endgroup$ Commented Aug 26, 2016 at 4:57
  • $\begingroup$ Yes I did miss that too. Will fix $\endgroup$ Commented Aug 26, 2016 at 4:58

1 Answer 1

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$$\sum_{i=1}^{n}{\frac{i}{(i+1)!}}=\sum_{i=1}^{n}{\frac{(i+1)-1}{(i+1)!}}=\sum_{i=1}^{n}\left({\frac{1}{i!}}-{\frac{1}{(i+1)!}}\right)=\frac{1}{1!}-\frac{1}{(n+1)!}$$

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