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?
The series does not converge

But if you treated it as a formal geometric series $1+r+r^2+\ldots= \dfrac{1}{1-r}$ and then let $r=-1$ you would get $\dfrac{1}{2}$ 

Similarly if $S=1-1+1-1+\cdots$ then you might set up 

 S = 1 - 1 + 1 - 1 + ...  
 S =     1 - 1 + 1 - ... 


and adding the two vertically gives

2S = 1
For divergent series, you can use Cesàro summation to find your desired limit.

The idea is to take the mean of your partial sums, and we hope it converges:

1 1 1/1
2 0 1/2
3 1 2/3
4 0 2/4
5 1 3/5


etc. you can see the mean goes to $1/2$.