This website hammers down on the first sum Numberphile uses, 1-1+1-1+...
This is called Grandi's series, look it up anywhere and you will find the answer IS 1/2, because of simple algebra.
People are always like "You can't use algebra on infinities", I have to respectfully disagree. In order to figure out infinities, you treat them like variables. The only difference between infinity and x is infinity can't equal something, x can. So when dealing with sums, we can treat infinity like a variable and figure out the sum from there. This is pretty much what Numberphile did the entire video excluding Grandi's series, which can be proven multiple ways elsewhere on the internet.