/sci/ - Science & Math

If [math] \sum_{n=0}^\infty [/math] is the notation for the sum in analysis, then [math] \sum_{n=0}^\infty n [/math] is divergent.

It can be pointer out that Peano arithmetic doesn't enable you to compute infinite sums, so e.g. to prove things like [math] \sum_{1=0}^\infty \frac{1}{2^n} = 2 [/math] or [math] \sum_{1=0}^\infty \frac{1}{n^2} = \frac{1}{6} \pi^2 [/math], you need uncountable sets such as [math] {\mathbb R} [/math] and a metric on it - i.e. I huge ballast of mathematical gadgetry.

There are other theories of infinite sums where the result are true, e.g. the Ramanujan sum. In the theory of (different) infinity sums, you can define properties your notion of sum should fulfill, and the sum over n is one that it's particularly hard to give finite meaning to. In comparison, for example, the sum over (-1)^n can be given a value in many theories of infinity sums.
Not all theories giving a value to n are algebraic in that sense, e.g. analytic continuation suggests a value for it too.
But again, the theory with most applications - analysis - doesn't assign a value to the sum over n.

-1/12 pops up in various related forms also in analysis, e.g. Lie theory and regularizations. The source is most often the second term in pic related.