[valtron]

There's only one way to add a finite list of numbers, but several ways to "add" infinite lists. Ramanujan summation is just one way. There are also other ways to define "addition" on infinite lists: Abel, Borel, Cesaro, Euler, etc. summation.

They're different from each other because they disagree on certain infinite sums, but they're still considered "summations" because, for finite lists, they give the same answer as normal addition.

[Zikes]

So the title should instead read "The sum of all positive numbers up to infinity (for a certain definition of 'sum') is -1/12".