The Numberphile youtube channel just
posted a video showing the extraordinary result that the sum of all natural numbers equals -1/12.
Of course many people would instinctively rather call it an outrageous claim instead, but seemingly there's some mathematical truth to the result, and there are many ways to prove it. (Note that this is not talking about what the limit of the series is, or what it converges towards, but rather the sum of an infinite amount of terms, which is different.)
Since one of the simple proofs shown in the video deals with infinities, I instinctively get the idea that
"you can prove anything with infinities" (in the same way as you can prove anything with something that equals eg. 0/0.) In other words, using some other mathematical trickery you could make the sum equal to whichever value you want, like -1/25 or 50.2. However, apparently -1/12 is the only result you get, no matter which method you use to solve it.
While the video (and its extra footage) has some talk about this, I would like to hear if someone else had some other insight into understanding the rather puzzling result.
(Also, why "12" and not something else? Where is precisely 12 coming from? I mean I can see the proof in the video, but it's still puzzling why its 12. Is there some kind of odd relation with precisely that fraction and the infinite sum of all natural numbers?)
