Suppose s is the infinite series 1 – 2 + 3 – 4 + …. The diagram above presents 4 copies of the series. Each white disk represents +1, and each red disk represents -1. Each pair of red and white disks annihilates to zero, and the connecting lines show that all of the disks beyond the first +1 (green) can be paired off in this way. The result is that 4s = 1 and, perversely, that 1 – 2 + 3 – 4 + … = 1/4.

(It’s not really that simple — this series doesn’t tend toward any finite limit, but any summation method that’s linear and stable does produce the sum 1/4.)