These mostly crop up in quantum field theory, where various formal expressions have infinite values. These can often be “regularized” to give finite results, or at least turned into a form that while still infinite, can be “renormalized” by such means as considering various terms as referring to observed values, rather than the “bare values”, which are carefully tweaked (often taking limits as they go to zero) in a coordinated way, so that the observed values remain okay.
Letting s be the sum above, in some sense what we’re “really” saying is that s = 1 + 2 s, which can be seen by formal manipulation. This has two solutions in the (one-point compactification of) the complex numbers: infinity, and −1. When doing things like summing Feynmann diagrams, we can have similar things where a physical propagator is essentially described as a bare propagator plus perturbative terms that should be written in terms of products of propagators, leading again to infinite series that diverge (several interlocked infinite series, actually—the photon propagator should include terms with each charged particle, the electron should include terms with photon intermediates, etc.).
IIRC, The Casimir effect can be explained by using Zeta function regularization to sum up contributions of an infinite number of vaccuum modes, though it is certainly not the only way to perform the calculation
Explicit physics calculations I do not have at the ready.
EDIT: please do not take the descriptions of the physics above too seriously. It’s not quite what people actually do, but it’s close enough to give some of the flavor.
I’m curious about this. More details please!
These mostly crop up in quantum field theory, where various formal expressions have infinite values. These can often be “regularized” to give finite results, or at least turned into a form that while still infinite, can be “renormalized” by such means as considering various terms as referring to observed values, rather than the “bare values”, which are carefully tweaked (often taking limits as they go to zero) in a coordinated way, so that the observed values remain okay.
Letting s be the sum above, in some sense what we’re “really” saying is that s = 1 + 2 s, which can be seen by formal manipulation. This has two solutions in the (one-point compactification of) the complex numbers: infinity, and −1. When doing things like summing Feynmann diagrams, we can have similar things where a physical propagator is essentially described as a bare propagator plus perturbative terms that should be written in terms of products of propagators, leading again to infinite series that diverge (several interlocked infinite series, actually—the photon propagator should include terms with each charged particle, the electron should include terms with photon intermediates, etc.).
IIRC, The Casimir effect can be explained by using Zeta function regularization to sum up contributions of an infinite number of vaccuum modes, though it is certainly not the only way to perform the calculation
http://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/ and the next two posts are a nice introduction to some of these methods.
Wikipedia has a fair number of examples:
http://en.wikipedia.org/wiki/1_−_2_%2B_3_−_4_%2B_·_·_·
http://en.wikipedia.org/wiki/1_−_2_%2B_4_−_8_%2B_·_·_·
http://en.wikipedia.org/wiki/1_%2B_1_%2B_1_%2B_1_%2B_·_·_·
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_·_·_·
Explicit physics calculations I do not have at the ready.
EDIT: please do not take the descriptions of the physics above too seriously. It’s not quite what people actually do, but it’s close enough to give some of the flavor.