>>12274699Given a sequence , the infinite sum in the theory of classical analysis is the number which fulfills
E.g. the number which fulfills
is .
That theory has a geometric model, i.e. it has some intuition and known applications.
There are other theories of infinite sum.
E.g. consider the (finite!) sum .
The theory of classical analysis doesn't assign a value to this sum, for every growing N.
The sum diverges, is what we say.
Now take a positive , then classical analysis (use the geometric series) proves
The limit of d to 0 of this sum is 1/2, but in analysis you can't switch the limits (because the limit is a ugly logical expression you saw above).
Other theories will straight out prove that the infinite sum is 1/2. Not standard theories you learn in school.
There are also regularizations of the initial finite sum (modifications of the original sum, usually with an extra parameter) depending on the bound N. Those (different!) regularized sums also converge to 1/2.
Quantum field theory does this all the time - adding extra dependent parameters on small, not yet observable scales to theories in hope convergence is improved.
As in, instead of trying the old to model physics, what's done is using a (usually more complicated) different thing and conjecturing that the world is actually better described by the regularized version of it.
