[reuben_tate]

Originally Posted by Zapmeister

e.g.

sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0

There's a lot of assumptions being made here and a lot of things done without justification. The terms in an infinite series are like the flesh of a cacti...they don't want to be ****ed with.

However, there is a lesson to be learned here. The lesson being that this way of defining how to given values to divergent series isn't "nice" and doesn't have the "nice" properties like the ones we see with convergent series (see this and this). We know with convergent series we can add term-by-term...but your thing shows that that is clearly not the case anymore (or else otherwise we get the absurdity that 0=1).

To be honest...this way of defining how to assign values to divergent series doesn't sit too well me either. It works well for complex analysts since analytic functions are "nice". However, this definition doesn't have a lot of the "nice" properties one is looking so if you find a "nicer" one let me know