1
$\begingroup$

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.

$\endgroup$
2
  • 3
    $\begingroup$ 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. $\endgroup$ Aug 26, 2016 at 4:57
  • $\begingroup$ Yes I did miss that too. Will fix $\endgroup$ Aug 26, 2016 at 4:58

1 Answer 1

14
$\begingroup$

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

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.