Fun with divergent series

A friend sent me a link to a Numberphile presentation in which Tony Padilla and Ed Copland show that the infinite series S = 1 + 2 + 3 + 4 + ... has (in some sense) the value - 1/12. It seems that this result is used in superstring theory in physics and goes all the way back to Euler in 1740. It has generated a lot of heat (and some light) on one of the math forums I follow, so I looked into it and have written it up a bit.

How is 1 + 2 + 3 + 4 + ... = -1/12 ?