12+34... = 1/4
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12+34... = 1/4
My friend brought this to my attention, and I've been in a sour mood ever since:
Here's the proof
As if that wasn't bad enough, using this result one can prove 1+2+3+4+... = 1/12
It's true!
Here's the proof
As if that wasn't bad enough, using this result one can prove 1+2+3+4+... = 1/12
It's true!
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 cmacis
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Isn't this one of Srinivasa Ramanujan's?
/me looks at wiki
No, it's Euler's. But naturally it fails at the divergency test. The terms in the sum don't tend to 0.
/me looks at wiki
No, it's Euler's. But naturally it fails at the divergency test. The terms in the sum don't tend to 0.
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Mathematician is a function mapping tea onto theorems. Sadly this function is irreversible.
QED is Latin for small empty box.
Ceci nâ€™est pas une [s]pipe[/s] signature.
 The LuigiManiac
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To show that the sum equals 1/4, they start out assuming that s = 1  2 + 3  4 + ... exists. Since it doesn't, their manipulations don't seem to be valid. I mean you could say:
Let x be infinity
infinity plus anything is just infinity
x + 3 is infinity
x + 9 is infinity
x + 3 = x + 9
3 = 9
or something along those lines.
Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
Let x be infinity
infinity plus anything is just infinity
x + 3 is infinity
x + 9 is infinity
x + 3 = x + 9
3 = 9
or something along those lines.
Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
 EradicateIV
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Infinite series has no meaning using ordinary arithmetic. You need to define it somehow. Usually you define it as the limit of partial sums, but you could use something else, eg Cesaro summation or Abel summation, to allow more series to converge.aguacate wrote:Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
 jestingrabbit
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The attempt, or desire or whatever, to give "valid" values to series and sequences which aren't convergent in the Cauchy sense is a well thought out area of mathematics. It disagrees strongly with the theory of Cauchy sequences, as you'd expect. Its all perfectly well defined and rigorous, its just not the theory you get from your undergrad calculus class.
Note that that working is under the heading "Heuristics for summation". Its a vague handwavy reason that Euler came up with. I'm sure that the first proof of the identity "1=0.9999..." similarly began with the same assumption. If you want to criticize it, read the stuff about Cesaro summation.
The series converges in the (C,2) sense. It doesn't converge in the Cauchy sense. Get over it.
aguacate wrote:To show that the sum equals 1/4, they start out assuming that s = 1  2 + 3  4 + ... exists.
Note that that working is under the heading "Heuristics for summation". Its a vague handwavy reason that Euler came up with. I'm sure that the first proof of the identity "1=0.9999..." similarly began with the same assumption. If you want to criticize it, read the stuff about Cesaro summation.
The series converges in the (C,2) sense. It doesn't converge in the Cauchy sense. Get over it.
aguacate wrote:Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
The result you are referring to requires a series which is already known to converge, but not to absolutely converge. So that result cannot be applied to "12+34+...", which does not converge at all.
Hix wrote:aguacate wrote:Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
The result you are referring to requires a series which is already known to converge, but not to absolutely converge. So that result cannot be applied to "12+34+...", which does not converge at all.
Well, I thought you might be able to do the same thing with this one, and hence why it worked, but after playing around with it for a bit, I figured out what it means, and even if you do rearrange the terms, the answer is still 1/4.
So the result 1/4 is a sort of average on the partial sums of the series 1  2 + 3  4 + ... You can see this by looking at the average of say the 51st term (26) and the 50th term (25) : 1/2. Then look at the average of the 52nd (26) and 51st term (26): 0. Now we compute the averages of the averages (1/2 and 0): 1/4. Voila! Very cool, I always thought there should be a way of dealing with divergent alternating series of this sort.
I still think it is misleading to say 1  2 + 3  4 + ... = 1/4. Of course this assumes that we all agree on what the three dots mean at the end of the LHS.
aguacate wrote:So the result 1/4 is a sort of average on the partial sums of the series 1  2 + 3  4 + ... You can see this by looking at the average of say the 51st term (26) and the 50th term (25) : 1/2. Then look at the average of the 52nd (26) and 51st term (26): 0. Now we compute the averages of the averages (1/2 and 0): 1/4.
wouldn't you get 0 if you average all of them?
 parallax
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Does anyone know of a proof that 1  2 + 3  4 + . . . still converges to 1/4 (in the Cesaro sense) even if you rearrange the terms?
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1  2) + (3  4) + (5  6) + . . . = 1 + 1 + 1 + . . . probably doesn't still sum to 1/4.
These are zetaregularized sums. I don't know if they are Cesarosummable.
1 + 1 + 1 + 1 + . . . = 1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = 1/12
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1  2) + (3  4) + (5  6) + . . . = 1 + 1 + 1 + . . . probably doesn't still sum to 1/4.
These are zetaregularized sums. I don't know if they are Cesarosummable.
1 + 1 + 1 + 1 + . . . = 1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = 1/12
hotaru wrote:aguacate wrote:So the result 1/4 is a sort of average on the partial sums of the series 1  2 + 3  4 + ... You can see this by looking at the average of say the 51st term (26) and the 50th term (25) : 1/2. Then look at the average of the 52nd (26) and 51st term (26): 0. Now we compute the averages of the averages (1/2 and 0): 1/4.
wouldn't you get 0 if you average all of them?
No, but you would approach zero as you average out n terms as n goes to infinity.
parallax wrote:Does anyone know of a proof that 1  2 + 3  4 + . . . still converges to 1/4 (in the Cesaro sense) even if you rearrange the terms?
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1  2) + (3  4) + (5  6) + . . . = 1 + 1 + 1 + . . . probably doesn't still sum to 1/4.
These are zetaregularized sums. I don't know if they are Cesarosummable.
1 + 1 + 1 + 1 + . . . = 1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = 1/12
Doesn't 1  2 + 3  4 + . . . diverge in the cesaro sense? I thought it was the Abel sum that gives it 1/4.
 jestingrabbit
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aguacate wrote:parallax wrote:Does anyone know of a proof that 1  2 + 3  4 + . . . still converges to 1/4 (in the Cesaro sense) even if you rearrange the terms?
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1  2) + (3  4) + (5  6) + . . . = 1 + 1 + 1 + . . . probably doesn't still sum to 1/4.
These are zetaregularized sums. I don't know if they are Cesarosummable.
1 + 1 + 1 + 1 + . . . = 1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = 1/12
Doesn't 1  2 + 3  4 + . . . diverge in the cesaro sense? I thought it was the Abel sum that gives it 1/4.
Its cesaro sum does indeed diverge. However, there is a family of generalisations, and it is (C,2) summable, in the sense outlined by wikipedia.
aguacate wrote:To show that the sum equals 1/4, they start out assuming that s = 1  2 + 3  4 + ... exists. Since it doesn't, their manipulations don't seem to be valid. I mean you could say:
Let x be infinity
infinity plus anything is just infinity
x + 3 is infinity
x + 9 is infinity
x + 3 = x + 9
3 = 9
or something along those lines.
Also, I know that nonabsolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1  2 + 3  4 + ...?
Only a little bit. The key here is that to make the manipulations rigorous one needs a more general defintion of convergence of a series. See for example http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation for a good intro of such methods and what they can give you. In these broader contexts one can make sense of series that one would otherwise label as divergent. Keep in mind that what we mean by convergence in these cases isn't exactly the same as what you are used to. But we can show for of the common generalized notion of convergence of a series one that when a series converges by both that notion and the conventional notion, the two will agree on the same number, and moreover any time the convential limit exists, the generalized one will also.

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The usual paradoxes you can come up with for summing conditionallyconverging sequences (or worse) involve permuting an infinite number of terms, or inserting an infinite number of brackets to force evaluation of an infinite number of subterms. That is, what fails is something like "infinite associativity" or "infinite commutativity"; but permuting or regrouping a finite number of terms is still okay, as is evalutating partial sums.
This can be used to motivate why the usual paradoxes which arise with the Grandi series can be dismissed:
1  1 + 1  1 + ... = 1  (1  1)  (1  1)  ... : forbidden, because infinitesummation is not infinitely associative.
1  1 + 1  1 + ... = 1  2 + 3  4 + ... : forbidden, because infinitesummation is not infinitely commutative.
1  1 + 1  1 + ... = 1  (1  1 + 1  1 + ...): okeydokey.
What I'm very uncomfortable with the treatment of the zetaregularaized sums. The fact that
1 + 1 + 1 + ... = 1/2
doesn't agree with
1 + 2 + 3 + ... = 1/12
is perfectly fine, from the POV that it equality would require the "infinite associativity" which we already know is not guaranteed, But it seems to me that the latter should at least be the square of the former. And the pseudoequation
11/12 = 1 + 1/12 = 1 + (1 + 1 + ...) = 1 + 1 + 1 + ... = 1/12
shows that you can't even split off a single term if you want welldefined results: evaluating expressions in terms of partial sums breaks things. Then, the meaning of any single addition is dependent on an infinite amount of context to its' left or right. That just ain't right.
Call these zeta continuations what you will, but calling it an infinite summation strikes me as pointless.
This can be used to motivate why the usual paradoxes which arise with the Grandi series can be dismissed:
1  1 + 1  1 + ... = 1  (1  1)  (1  1)  ... : forbidden, because infinitesummation is not infinitely associative.
1  1 + 1  1 + ... = 1  2 + 3  4 + ... : forbidden, because infinitesummation is not infinitely commutative.
1  1 + 1  1 + ... = 1  (1  1 + 1  1 + ...): okeydokey.
What I'm very uncomfortable with the treatment of the zetaregularaized sums. The fact that
1 + 1 + 1 + ... = 1/2
doesn't agree with
1 + 2 + 3 + ... = 1/12
is perfectly fine, from the POV that it equality would require the "infinite associativity" which we already know is not guaranteed, But it seems to me that the latter should at least be the square of the former. And the pseudoequation
11/12 = 1 + 1/12 = 1 + (1 + 1 + ...) = 1 + 1 + 1 + ... = 1/12
shows that you can't even split off a single term if you want welldefined results: evaluating expressions in terms of partial sums breaks things. Then, the meaning of any single addition is dependent on an infinite amount of context to its' left or right. That just ain't right.
Call these zeta continuations what you will, but calling it an infinite summation strikes me as pointless.
 jestingrabbit
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infinigesimal wrote:What I'm very uncomfortable with the treatment of the zetaregularaized sums. The fact that
1 + 1 + 1 + ... = 1/2
doesn't agree with
1 + 2 + 3 + ... = 1/12
is perfectly fine, from the POV that it equality would require the "infinite associativity" which we already know is not guaranteed, But it seems to me that the latter should at least be the square of the former. And the pseudoequation
11/12 = 1 + 1/12 = 1 + (1 + 1 + ...) = 1 + 1 + 1 + ... = 1/12
shows that you can't even split off a single term if you want welldefined results: evaluating expressions in terms of partial sums breaks things. Then, the meaning of any single addition is dependent on an infinite amount of context to its' left or right. That just ain't right.
Call these zeta continuations what you will, but calling it an infinite summation strikes me as pointless.
Agreed. The Cesaro theory makes a lot of sense to me, but using analytic continuations to come up with values of sums seems entirely wrong. For instance, if all the terms are positive, then the sum ought to be positive too. Linearity is a must imo, and correct behaviour wrt Cauchy products is a big plus. Cesaro summation has all this.
Euler was notorious for proofs of this sort. Keep in mind he worked a little bit before the development of calculus, and his lack of rigor or concern for convergence when working with series was one reason calculus was welcomed with such reluctance and suspicion. However, when he wrote "..." he implicitly meant a finite sum instead of an infinite one. Can't remember the example; will look it up later.
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archgoon wrote:Can anyone give me a somewhat natural example of how a cesaro sum might occur in solving a problem? Preferably the example would involve "wanting to sum a series" but not being able to unless you cesaro sum.
According to Wikipedia, Grandi proved that his series (11+11+11+1...) should converge to 1/2 through geometric considerations. Unfortunately, it gives no hint of what these are. In general, the "sums" attached to nonconvergent series seem to come up in geometry or even physics, but I have no idea how. But it does seem that attaching sums to such series is not entirely pointless. More details can be found at History of Grandi's series (Wikipedia).
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 jestingrabbit
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archgoon wrote:Can anyone give me a somewhat natural example of how a cesaro sum might occur in solving a problem? Preferably the example would involve "wanting to sum a series" but not being able to unless you cesaro sum.
Most of the problems it solves are theoretical in nature. Its used extensively in ergodic theory. Its also used in some Fourier stuff. Its not like there's an easy example of where its needed. Its "nice" in that it explains Grandi's and Euler's intuitive proofs, but its not needed for much else that isn't largely theoretical.
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