The Ramanujan sum of all positive numbers up to infinity is -1/12
kottke.org
kottke.org
Now, I'm no mathematician, but I'm betting that (R) is important, and affects the true value of the -1/12, and probably should not be omitted.
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.
The sum of alternating +1 and -1 equals 1 if you stop on an odd, and 0 if you stop on an even. Infinity is treated differently in different fields. I suppose in physics, it's probably treated as a superposition of even and odd, so statistically, you could interpret it as 0.5. And I'm sure that works great for physicists and it fits a number of scientific models. But there are probably other fields involving mathematics that would find this interpretation to be silly and not very useful.
The Ramanujan sum of all positive numbers up to infinity is -1/12
Which is not sensational, exciting, or even that interesting.
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
...so the way we normally think of sums, of course 1 + 2 + 3 ...grows infinitely large. Makes me wonder- what was the impetus for creating the idea of Ramanujan sums?
It is only true if you redefine what the "sum" of an infinite series is.
Well first, Achilles has to travel 1 unit because Tortoise got a head start. That takes 1s. By then, Tortoise managed to run x units, so Achilles has to catch-up, which takes another x seconds, and by then... etc: The time it takes Achilles to catch up is 1 + x + x^2 + ... Since x < 1, this converges and the answer is 1/(1-x). It takes 1/(1-x) seconds for Achilles to catch up.
Plot the lines: Achilles runs along A(t) = t, the tortoise T(t) = 1 + x t. Solving for t: t = 1 + x t = t (1 - x) = 1; t = 1/(1-x).
Ok, now what if x > 1? Let's use x = 2.
1/(1-2) = -1. It takes -1 seconds for Achilles to catch up. Don't believe me? Plot it: A(t) = t, T(t) = 1 + 2t; Achilles catches up with Tortoise 1 second before the race starts!
There you have it, a physical interpretation of why 1 + 2 + 4 ... = -1. (I have no example right now for 1 + 2 + 3 ..., sorry.) So you see, this isn't nonsense, though I admit it DOES depend on definitions!
Either I'm a really grumpy person, or their videos are actually terrible. I think there are enough interesting things in math that can be explained without falsely embellishing uninteresting results and ideas.
Like what has already been said I think this is helpful in certain fields, but bollocks in others.
http://www.reddit.com/r/math/comments/1usu93/1_2_3_4_5_112_n...
Reddit has major issues due to its user base, but one thing I do is browse certain subreddits once a week (month) by filtering by top posts in the past week (month). The day by day drone in most subreddits is tiring, but quality threads hit the top posts.
I suspect this kind of reasoning is useful in some fields for some particular kinds of infinite sums, but for the particular summation problem as stated, as most people understand it, this is rubbish.
Summary: as most people understand it, the sum of all natural numbers is (a) positive (b) very large (c) not finite (d) certainly not -1/12.