Phil Plait in Slate:
There have been a lot of objections raised about the video mentioned below and the way I describe the math therein. It's more than I can simply add or update here, so I'm going over all the input and will write a follow-up soon. Very briefly: The series manipulation done in the video is a bit of a trick and is not rigorous, but there are rigorous mathematical solutions which do work under certain circumstances. However, interpreting them is tricky. Again, more coming soon.
This post involves math. Really bizarre, brain-melting math. The math itself is actually not that complicated—I promise!—but the result will carve out a piece of your soul and leave hollow space. Seriously, please, bear with this. I pounded my head on my desk enough to leave a dent in the wood and a welt in my forehead to figure this out, so please just stick with it. When it’s done, you’ll have a better appreciation for just how completely flippin’ weird our Universe is.
Math can be a bit tricky sometimes.
OK, stop your uproarious laughter. I know, most normal people cringe in fear when math pops up in their lives. Of course it can be a bit tricky. But I’m not talking about word problems, or algebra, or trying to figure out compound interest (or balancing a checkbook, my own personal unscalable mountain).
I mean math itself. You can sit down and write out a few simple things and wind up with an answer so bizarre, so counterintuitive, that you figure it must be wrong.
More here. And a rebuttal here. I have asked a few math/physics friends to shed further light on this. If and when they do, I will give their responses here.