Expand the summation (university)

Question

Expand the summation $$\sum^{n}_{i=0}i\times i! = $$

I am studying for an exam, I have no idea what this question means.

Answer 1

Let $$\displaystyle \bf{S_{n} = \sum^{n}_{i=0}i!\times i = \sum^{n}_{i=0}i!\times [(i+1)-1]} = \sum^{n}_{i=0}\left[i!\times (i+1)-i!\right]$$

So we get $$\displaystyle \bf{S_{n} = \sum^{n}_{i=0}\left[(i+1)!-i!\right]}$$

Now Expanding Summation or Using Telescopic Sum , we get

$$\displaystyle \bf{S_{n} = \sum^{n}_{i=0}\left[(i+1)!-i!\right]} = (n+1)!-1$$