1+2+3+4+5+... = -1/12
22.214.171.124.5... = sqrt(2π)
Before you think I've lost my sanity, these results are examples of zeta regularisations which are frequently used in theoretical physics to 'renormalise' the infinite sums that have a habit of appearing there. Although their use in physics isn't terribly well justified, the zeta regularisation, due to Hawking, is well defined as a mathematical operation.
Methods for finding 'sums' of divergent series have a long pedigree. In fact, Hardy wrote an entire book on the subject. The techniques he uses range from Cesaro means to Borel summation.
Zeta regularised sums arise in the selection of the 'critical' dimensions in which various string theories work. For example the fact that bosonic string theory works in 26 dimensions can be seen as coming from 1+2+3+...=-1/12. Coming back down to Earth - a similar computation arises when computing the Casimir force, something that's actually measurable in a lab.