>>12542089>For one, how can you add an infinite number of numbers? If you accept the real number line, then you can define a metric on it and use it to give implicit definitions of infinite sums.
The standard theory, anaysis, proves infinite sums of rationals converging to real numbers, e.g.
or
It also proves
These have nice geometric and graphical meaning attached to them.
Other more exotic summation methods of summation may give
But those theories are much less useful and, in turn, less studied.
>Secondly, how could adding positive integers ever get you a negative number?The standard formalization of natural numbers, Peano arithmetic, can't make claims about infinite sums.
If you believe in the real number line (or if you're just a formalist), then real analysis easily proves that the limit of
converges to 0 as m goes to zero, so this more common theory doesn't predict a negative sum.
