-1/12

http://www.smithsonianmag.com/...234-112-180949559/?no-ist

http://physicsbuzz.physicscentral.com/...4-112-absolutely-not.html

in normal mathematics generally infinity - infinity is undefined so you can't use subtraction to cancel them out. There are various types of infinity which are of varying sizes and in the physics case there is a specific definition/instance where it makes sense to do so, but that's not true in general and not for the series in general.

Numbers aren't real things. Numbers are symbols that we use to assign a value to something so that we can manipulate it in the real world. Divergent series are not real, tangible things that we can manipulate in real life. You can't pick up 1 thing, then 2 things, then 3 things, then 4 things... etc. and count them all. So this is very simple. In order to manipulate and understand the real world, mathematicians and physicists had to assign a value (-1/12) to the sum of all natural numbers. That doesn't mean that anybody actually believes that if you pick up one thing, then two, then three, etc. in the end you'll have -1/12 things. It's just a symbol used to turn this theoretical, fake, abstract concept (an infinite series) and use it in real world applications. It always amuses me when people get all up in arms about this. Numbers are tools that we use to count things. Nobody is suggesting that we can actually count infinity.

@hawkeye: math isn't science, it's a well defined system of logic (and is closer to philosophy). It is a powerful tool used to help explain the world around us and accomplish a large number of tasks and deal with abstract concepts.
It's more than simply "counting" objects. There are several inventions that you use regularly that discredit your assertion that divergent series are not real tangible things that you can manipulate in real life, also several observable phenomena. By the logic rules of mathematics, -1/12 is not a valid conclusion in general. We can redefine what "infinity" and addition means so that it does, but we seek a coherent set of rules.

Divergent series are more useful as a predictive tool to indicate when a process is going to fail/break versus not.
You can't "count" to infinity. Infinity is not part of any number set and is a separate concept in mathematics. A lot of statistics deals with the concept of convergence (various kinds) and infinity. Of course statisticians *know* that you don't have an infinite sample or whatever, but it is more like a direction post to show you what happens when problems get larger. How you address problems that initially follow divergent series is important if you want to come up with something meaningful.


Humans have no grasp on reality anyway because we can only perceive things as they were (speed of light/sound) and in a limited sense. Mathematics helps speed up the scientific process as it can give us insite into the unobservable and point us in the right direction. Our lack of advancement in some areas of genetics is due to missing gaps in our mathematical and physical knowledge. We lack the means to form a meaningful hypothesis (for now). For example: if I give you a desired protein shape, can we design a DNA sequence from scratch that will make that protein? Not with any degree of complexity..

"There are several inventions that you use regularly that discredit your assertion that divergent series are not real tangible things that you can manipulate in real life, also several observable phenomena. "

Tell me more.

There are a number of scientific proofs of these weird sums in nature. Case and point, the Casimir Effect.

When it was theorized it had the sum 1 + 2 - 3 + 4 .... to infinity in it as part of a constant. When you do the sum it comes out to 1/4. Well fast forward 50 years and somebody finally is able to measure it and that 1/4 in seen in the actual measurement.

100% of what I know about this stuff is from Numberphile :P

pretty sure assigning values like 1/2 to 1 -1 +1 -1 ... is not exactly ok, the series is ever only 1 or 0 and since it alternates it is technically divergent so when they use that result it goes down a path where anything they derive isnt really true.

plus i think someone posted a link but 1 + 2 + 3 + ... = -1/12 *plus infinity*. ignore the infinity part and you have your answer. basically the link said you could use a generating function using gauss' method for adding up a series of integers from 1 to n and apply the function over (-infinity, infinity). you get a parabola and the integral of the positive left side = integral of positive right side but it dips under the x-axis a bit for an area of -1/12. so the integral over [0,infinity) IS infinite, obviously, but slightly (-1/12) smaller than the integral of the generating function over (-infinity,0].

what I gather is that the series and -1/12 are related but not equal, as your intuition would expect.

the -1/12 thing only works for a certain type of infinity which is used in some cases in physics. But there are infinite types of infinity (as proven by Cantor). When using the concept in math in general it has to satisfy infinity over all types. Harmonic series where the absolute value of the terms do not get successively smaller do *not* converge on this basis. When we use the "infinity" symbole in most cases we really mean limit as x approaches infinity, especially in real analysis. This is not adding infinite terms, but rather what result the function/series/whatever gets closer and closer to the larger it gets.

1-1+1-1+1.... is not divergent, it's undefined. There's a huge difference. Just like what happens in lim x->infinity of sin (x). It doesn't get successfully larger, but it also doesn't get closer to any particular answer.

For those of you that know about statics, there are different kinds of convergeance. For example if u is a uniform random variable between 0 and 1, then 1 - u has the same distribution (converges in distribution). (Functionally they converge). However it doesn't converge in probability or absolutely. While for simulation purposes this really doesn't matter, for other applications it certainly can.

hawkeye: signal processing and fluid dynamics are examples.

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0/0

What have we done

we sit by and watch the world implode

quiet sean