Note that n*(n-1)! = n! so if you simplify the denominator (in terms of it being a factorial) then see if you get a cancellation.
I'm not very familiar with how to post the expression here as it is, but this is a link to an entry for it on WolframAlpha: sum from i=1 to n of (-1)^(i+1)*(i)*& #40;n+1)!*1/((i+1& #41;!*(n-i)!) - Wolfram|Alpha
I have also attached an image that shows what the series is. It also shows that the answer is equal to 1. However, Wolfram doesn't show any steps and despite my attempts, I haven't been able to prove that. I would rather not have the entire answer spoiled, but any help would be appreciated.
Binomial coefficient - Wikipedia, the free encyclopedia
There are some in that page, but I haven't been able to put any of them to good use. :S