How Does Physics Deal with Infinity?

There's been some discussion about the video where I claimed that 1+2+4+8+16 and so on is equal to negative one - apparently I'm on my way to math jail. But actually, adding up divergent series is an old topic in math, and more importantly, it's actually relevant to physics! In physics, there are a lot of hard problems where we can't find the whole answer - but we can find the answer to a simplified version of the problem, and then correct it step by step so we get closer and closer to the real answer. Check it out: we can approximate this hyperbola by starting with a line, adding a curve, a wiggle, and onwards… except that our step-by-step process only works for a little part of the curve! Outside this "area of convergence", we're totally wrong! Ok, back to physics - Some problems, like figuring out how electrons scatter in quantum field theory, are really hard to solve! But we know there must be an answer, because we do experiments and see that something happens. So we take the step-by-step approach - except that most of the time, the corrections get bigger and bigger, adding up to infinity! What went wrong? We're outside the 'area of convergence'. But luckily, all is not lost! Because, just as the series 1+2+4+8+16 etc. contains enough information to tell us that it really means negative one, physicists use clever techniques to tease the real answer out of infinity. For example, they've calculated quantities, like the strength of the electromagnetic force, to incredible detail - and the calculations agree with experiment! This is one of the most accurate and precise results in all of science – and that's the power of taming infinity.