Any sum sequence can be split up into its components and re-added, because sum functions are not signs that are separate, but notations describing a number itself. For example, 1-1 can be written as (+1) + (-1). Having proved this, we know any sum can be broken into its sum parts and still result in the same answer. (-1) + (+1) = 1-1. Therefore, any series can be broken down into its components, provided we know all of the components. For "S = 1-1+1-1+1-1+1-1...", we know that there are an equally infinite number of (+1) and (-1). The proof is thus: One single infinite sequence equals one (1) infinity. Therefore, two infinite sequences of the same type equals two (2) infinities. Having proven thus far that any sum can be broken down into its sum parts, AND that any infinite sequence containing two infinite sequences of the same type is even, we can break down "S = 1-1+1-1+1-1+1-1..." into "S = (-1-1-1-1...) + (+1+1+1+1...)". We know (-1-1-1-1...) tends towards negative infinity. Conversely, (+1+1+1+1) tends towards positive infinity. These two infinities of the same length cancel each other out; (-?) + (+?) = 0. Thus, "S = 1-1+1-1+1-1+1-1... = 0", or "S = 0".

1-1+1-1+1-1+1-1... is equal to zero (0)