Euler was tricksy, he was.
We all know and love geometric series. It’s the way that we know that 1 + 1/2 + 1/4 + 1/8 + 1/16 + … = 2. In general, if you have a^n, then the sum as n goes to infinity is going to be 1/(1-a). Euler used this to show that
1 + 2 + 4 + 8 + 16 + 32 + … = -1
Like I said, tricksy. Unfortunately, unlike a bunch of the awesome unlikely stuff he said that turned out to be right, this one doesn’t work so well outside of some very odd frames of reference.
But why doesn’t it work? There’s an Official FTG Not Remotely A Prize for those who e-mail the answer to before October 29, 2007, so drag out your copy of Stewart’s Early Transcendentals and lose yourself in the beauty of pure math for a while!
– Count Dolby von Luckner