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THE SUM OF ALL NATURAL NUMBERS
1 + 2 + 3 + 4 + 5 + ... → ∞ = ?
Let S stand for Series.
S = 1 + 2 + 3 + 4 + 5 + ... → ∞
S1 = 1 - 1 + 1 - 1 + 1 - ... → ∞
S2 = 1 - 2 + 3 - 4 + 5 - ... → ∞
S1 = (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 ... = 0
S1 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) ... = 1 + 0 + 0 + 0 + ... = 1
1 - S1 = 1 - (1 - 1 + 1 - 1 + 1 - ... ) = 1 - 1 + 1 - 1 + 1 - 1 + ... = S1
1 - S1 = S1
1 = 2 S1
S1 = 1/2
S2 = 1 - 2 + 3 - 4 + 5 - ...
2S2 = (1 - 2 + 3 - 4 + 5 - ...)
+ (0 + 1 - 2 + 3 - 4 + 5 - ...) = 1 - 1 + 1 - 1 + 1 ... = S1
2S2 = S1 = 1/2
S2 = 1/4
S - S2 = (1 + 2 + 3 + 4 + 5 - ... )
- (1 - 2 + 3 - 4 + 5 - ... )
= 0 + 4 + 0 + 8 + 0 + 16 + .... = 4 (1 + 2 + 3 + 4 + ...) = 4S
S - S2 = 4S
S - 1/4 = 4S
- 1/4 = 3S
Marco Chimienti
14.12.16
1. Here’s what just happened: we’ve added every positive intereger, every single one up to infinity, and arrived at -1/12. Some steps were skipped, but nothing on this page is invalid. There is one step which feels like you’re braking some rules, but I’ve been assured it’s valid. In fact, this is what’s called a divergent series, and using some complex maths, we can assign a value to numbers which, intuitively, would either explode and reach infinity, or oscilate between values, as in S1. So what does this mean? I don’t know. I’m not a mathematician. Maybe it means there’s something up with infinity. Maybe it means nothing — just a glitch in the rules.
I was browsing the stacks at BMV. I had already found a book by Nabokov (Pale Fire), which wasn't the one I'd been hoping for (Lolita), but it would do. I ended up in the 'W' section looking under Wallace, always hoping there'd be something new, some previously unpublished work. Instead, I was treated to a copy of Everything and More - a book I wasn't even aware he'd written - about the mathematical concept of infinity (of all things). That my main stylistic and moral guide throughout this process also wrote about infinity, saw that same beauty, came as a quite a pleasant surprise. I wonder now if I should revisit a subject I had set aside, labeled as a 'dead end'. There are times when I'm certain this thesis is really about beauty, and that's a problem, because that means everything is fair game.