0 or 1?

[casrip1@gmx.com]

saw this very interesting video:

and wanted to share/talk about it in a place without 500 character limit lol

 

i personally believe the answer can be considered undetermined, or both 0 and 1.

 

to have a definite answer, we gotta know the number of 1's we have, and the operation sign we start with. starting with a - sign and having odd number of 1's means you have the final answer = 1 (1 - 1 + 1 = 1) and having even number of 1's means you have final answer = 0 (1 - 1 + 1 - 1 = 0).

 

infinite, by oxford's definition means "impossible to measure or calculate" which means there is an undetermined numbers of 1's involved. to want an definite answer is to collapse the infinity and assign it a finite numbers of 1's (either an odd or an even number) which would give you the answer of 1 or 0 respectively (assuming starting operation is - ). so the answer can be assumed to exist in a superposition of both 0 and 1 and it will randomly result in either 1 or 0 depending on the number of 1's once the infinity is collapsed.

sound familiar? (Schrodinger's cat)

[D H]

As everyone knows, 1+1+1+... = -1/2.

 

How to deal with diverging series is a bit challenging. The easiest answer is to say that diverging series don't have a sum. After all, failing to converge to a sum is exactly what diverging means.

 

A more complex answer is that divergent series such as 1-1+1-1+..., 1+1+1+..., 1+2+3+4+..., 1+2+4+8+... can reasonably be given an answer by means of analytic continuation. Analytic continuation is a very powerful tool. It won't just give an answer to these seemingly nonsense sums, it gives a unique answer. With this approach, the values for these series are 1/2, -1/2, -1/12, and -1, respectively.

 

Consider the expression [imath]\frac 1{1-x}[/imath]. This expression has a value for all x except for 1. This expression can be expressed as an infinite series, [imath]\frac 1 {1-x} = 1+x+x^2+\cdot = \sum_{n=0}^{\infty} x^n[/imath] provided that the series converges. This series has a radius of convergence of 1. What if I naively plug x=-1 into both the expression and the series? The series that results is Grandi's series, and the expression says the value should be 1/2. That's the same value as that obtained by averaging and also by the neat little trick of finding that 1-S=S. Note that all of these "tricks" yield the same value, 1/2. Maybe there's more to this than just a trick. That's just what Hardy saw in his book Divergent Series. The reason all of these tricks yield the same answer is because of some deep properties of analytic continuation.

[casrip1@gmx.com]

i'm sorry, but did you just state 1 + 1 + 1 + ... = -1/2 and 1 + 2 + 3 + 4 + ... = -1/12 and 1 + 2 + 4 + 8 + ... = -1? could you please elaborate how an infinite sum gave you an answer smaller than the smallest number in the series (1)? and try to use terminology understandable by a mediocre high school student. and as for the 3rd paragraph, i understood about as much as an ant (assumingly) understands about the vastness of the universe

[mathematic]

Example: 1 - x + x^2 - x^3 + ..... = 1/(1+x) for |x| < 1. (singularity at x = -1). Analytic continuation to all x (except -1). Plug in x = 1 and get 1 - 1 + 1 - 1 .... = 1/2. Plug in x = -2 and get 1 + 2 + 4 + 8 +... = -1/2.

[D H]

All of these sums are examples of infinite whose partial sums don't converge to a single finite value. They are divergent series. In high school and lower level college math course you are taught that those divergent series don't have a value. They diverge, after all. That hasn't stopped mathematicians from trying to give meaning to divergent series. A number of different techniques have been developed. The averaging approach used by Grandi is one of the simpler techniques. This approach works nicely on series that are bounded but nonetheless fail to converge to a single value. Here the partial sums either alternate between a finite number of finite values or converge toward a set of alternating values. This averaging approach won't work on a series such as 1+2+4+8+... because the partial sums diverge rather than alternate.

 

The way to assign a meaningful value to those divergent series that truly are divergent is via analytic continuation. The basic idea is to find an analytic function whose domain is larger than the series' radius of curvature and whose value is equal to that of the series over the series' radius of curvature. In a sense, that analytic function provides meaning to the series even outside the series radius of convergence.

[daniton]

Example: 1 - x + x^2 - x^3 + ..... = 1/(1+x) for |x| < 1. (singularity at x = -1). Analytic continuation to all x (except -1). Plug in x = 1 and get 1 - 1 + 1 - 1 .... = 1/2. Plug in x = -2 and get 1 + 2 + 4 + 8 +... = -1/2.

Since you restrict the function for |x| < 1 why do you plug number not from the domain and also when we generalize binomial theorm to binomial series we use the same restriction so that the function makes sense like when you plug 0.

[casrip1@gmx.com]

so you're saying 1/(1+x) where x is one of these series is what you are using to get the answer for the series? if so then thats not really the answer for the series, thats the answer of 1/(1+series). i meant for a discussion about the series itself, not some function of it

[mathematic]

Since you restrict the function for |x| < 1 why do you plug number not from the domain and also when we generalize binomial theorm to binomial series we use the same restriction so that the function makes sense like when you plug 0.

That is the result of analytic continuation. The closed form of the series can be extended to all x, except x = -1, even though the series itself cannot.

[D H]

so you're saying 1/(1+x) where x is one of these series is what you are using to get the answer for the series? if so then thats not really the answer for the series, thats the answer of 1/(1+series). i meant for a discussion about the series itself, not some function of it

 

That is not what I'm saying. Expand 1/(1+x) as a series. You'll get 1-x+x²-x³+... or [imath]\displaystyle{\sum_{n=0}^{\infty} (-1)^n x^n}[/imath]. Plug in x=1/2 and the series becomes 1+1/2+1/4+1/8+..., which is 2. The left hand side yields 1/(1-1/2), or 2. The series and the generating function agree for x=-1/2. This is true for all x between but not including -1 and 1. It's also true for all complex numbers with ||x||<1. This series has a radius of convergence of 1.

 

Let's ignore this minor inconvenience and set x to 1. You'll get 1-1+1-1+... : Grandi's series. The generating function yields a value of 1/2. So in a sense, 1-1+1-1+... = 1/2. Setting x to -2 yields the series 1+2+4+8+... Here the generating function yields 1/(1-2)=-1. So in a sense, 1+2+4+8+... = -1.

 

How valid is this? What you will be taught when you study series in high school, again in freshman calculus, and yet again in complex analysis says that what I just did is utter rubbish. There are however a number of reasons that say that this particular abuse is valid. The math that says this is valid is beyond what you'll learn when you're a sophomore or junior in college.

 

This abuse of mathematics (or neat extension to mathematics) is critical in higher level quantum mechanics such as quantum field theory. The neat thing is that the experimental outcomes are consistent with what this renormalization nonsense says should be seen in those experiments.

[mathematic]

Analytic continuation is a concept introduced in an advanced course (complex variables) - senior college or graduate school.

 

For high school or college freshman it should be ignored.

[md65536]

I'm no expert.

 

When thinking of adding up these series as a process, it is a process that never ends. If the series converges, it can be considered a process that "ends after an infinite number of steps".

 

"Never" here refers to the number of steps. If you map each term to a length of time, and the series adds up to a finite sum, then it can end after a finite amount of time. Eg. Zeno's paradox. It's very much the same thing: A process that can only end after an infinite number of steps can also end after a finite amount of time if you can perform an infinite number of steps in a given time.

 

But what we must carefully avoid is attaching incorrect meaning to either the numbers or the process of summing them. Numbers and math are abstractions, that obey certain rules, and things that have the same properties can also obey the same rules, but not everything has the same properties. These numbers are not physical things that exist. To say that a non-converging series is "equal" to a single value, has as much meaning as you give it, and is only true for things that share that meaning. For example in OP's linked video, they map the summing of 1 - 1 + 1 - 1 + ... to the process of turning a light on and off, and the duration between switches to 1, 1/2, 1/4, ... This means that the rate of switching approaches infinite for a moment. Is that meaningful? There's no answer to what the state of the light is after an infinite number of switches in a finite number of time if it's not physically possible to do that. The series, and the elusive single "answer", is applied incorrectly to something that doesn't have the same properties.

 

The answer of "the process doesn't end and doesn't have an answer", and the different result of the analytic continuation and its answer, each appropriately apply to different things. If you're thinking about it as if numbers have a physical existence independent of whatever they might represent, and there must be a single answer to every equation, then I think you'll end up stuck with paradoxes due to having made bad assumptions.

Edited July 3, 2013 by md65536

[phyti]

U=1-1+1-1+…

(-1)*u+1=u

2u=1

u=1/2

The series oscillates between 0 and1,

like a superposition of states,

or like a tossed coin before it comes to rest.

It has both values with an average,

(contrary to the minds oversimplistic expectation that it should have one value).

Find a good book on fuzzy logic, it may help.

 

[studiot]

 

Analytic continuation is a concept introduced in an advanced course (complex variables) - senior college or graduate school.

 

Analytic continuation does not only apply to complex analysis. It is a very useful method that we rely on in numerical methods, differential geometry, real analysis, fourier analysis and and really anywhere that we want to include points on the the domain boundary in our analysis.