My "factorial" abilities are a slightly rusty and although I know of a few simplifications such as: $(n+1)\,n! = (n+1)!$, I'm stuck
I have to prove by induction that:
$$\sum_{i=1}^n\frac{i-1}{i!} = \frac{n!-1}{n!}$$
I get so far as: $$\frac{k!-1}{k!} + \frac{(k+1)-1}{(k+1)!} = \frac{(k+1)!(k!-1) + k\cdot k!}{k!(k+1)!}$$
and I know I should get: $$\frac{(k+1)! -1}{(k+1)!}.$$
I don't see how, though. Any help would be appreciated, thanks!
Hint: Instead of taking $k!(k+1)!$ as the common demoninator, simply take $(k+1)!$ as the common denominator. Then $$\frac{k!-1}{k!}+\frac{(k+1)-1}{(k+1)!}=\frac{k!-1}{k!}+\frac{k}{(k+1)!}=\frac{(k!-1)(k+1)}{(k+1)!}+\frac{k}{(k+1)!}.$$ Can you take it from there?