power series - How can we know the answer to 1-1+1-1+1...?

I was watching this video I noticed that the teacher said that 1-1+1-1+1... equals 1/2. How can we know that? The proof he uses doesn't make sense to me. We go from 1 to 0 to 1 and back again, etc. If it goes on like so forever, where does a fraction come into play?