Bad math

For the discussion of math. Duh.

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Re: Bad math

Postby jestingrabbit » Sat May 24, 2014 8:49 am UTC

Yeah, if you don't get up to the Cauchy integral theorem/formula, its not a real complex analysis course imo. That said, we had one of those in second year.

How does your uni train electrical engineers if you don't force them to do serious complex analysis?
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Re: Bad math

Postby mike-l » Sat May 24, 2014 3:47 pm UTC

Spambot5546 wrote:
mike-l wrote:Sometimes I feel like the folks who get really upset about 1+2+3+.... = -1/12 are those who know just enough math to be completely uninteresting to talk to.

You're talking about the principal like anyone who doesn't know about it doesn't know much math. That might be true at a post graduate level, but one can know a lot of interesting math without ever having learned this particular esoteric version of summation.

Again, I didn't say anything about not knowing much math.

What I was saying is that the argument I was originally quoting, as well as other complaints about the numberphile video, feel to me like they are coming from someone who knows some amount of mathematics and has the arrogance to think that if they don't understand an idea it must be bunk. The point has nothing to do with how much you know and everything to do with how you deal with things you don't.
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Re: Bad math

Postby gmalivuk » Sun May 25, 2014 12:49 am UTC

Spambot5546 wrote:Did you guys seriously get to take complex analysis classes as undergrads? My school considered real analysis a 400 level, and the only class dealing with the complex plane was Complex Variables, the 300 level that introduced the concept. Did I go to a shitty university? :'(
Honestly? I had actually written a post that pretty much speculated that you did, in fact, go to a university with perhaps a somewhat less comprehensive math program than one might hope.

Isn't 400-level still undergrad? As I recall one of the times I remember seeing other summation methods was in Math 451 at my university, which was (almost?) entirely populated by undergrads.
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Re: Bad math

Postby Eebster the Great » Sun May 25, 2014 1:44 am UTC

gmalivuk wrote:Isn't 400-level still undergrad? As I recall one of the times I remember seeing other summation methods was in Math 451 at my university, which was (almost?) entirely populated by undergrads.

Yeah, but if real analysis is 400-level, then complex analysis should be graduate level.

Also, in the courselist you linked, analysis is 500-level.

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Re: Bad math

Postby gmalivuk » Sun May 25, 2014 4:19 am UTC

Eebster the Great wrote:
gmalivuk wrote:Isn't 400-level still undergrad? As I recall one of the times I remember seeing other summation methods was in Math 451 at my university, which was (almost?) entirely populated by undergrads.

Yeah, but if real analysis is 400-level, then complex analysis should be graduate level.

Also, in the courselist you linked, analysis is 500-level.
Yes, the alpha-level courses on that undergraduate list that actually have "analysis" in the names are 500-level (though complex is first and they're expected to be taken in one year, because you don't need to take one class two semesters in a row). They were still around half-filled with undergrads, though, because they aren't actually grad courses so much as grad-prep courses. (As i recall, I took 512, 513, 590, and 595 as an undergrad.)

But in any case, what I said was that I learned about other summation metods in 451,which is of course a 400-level class.
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Re: Bad math

Postby mike-l » Sun May 25, 2014 2:01 pm UTC

At my school the first analysis course was in the second half of second year, followed by third year courses in both real and complex analysis. 4th year courses were largely specialized subjects and/or cross listed grad courses. I'm not nearly as familiar with the undergrad curriculum at my grad school, but I do know there was a third year real analysis course, as my advisor taught it and I would fill in for him on occassion
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Re: Bad math

Postby Twistar » Sun May 25, 2014 3:06 pm UTC

I didn't have to take complex analysis to get my math BA. in fact I avoided it because I was scared! I took stuff like PDE's and Fourier analysis instead. I had about 1/2 of a lecture on cesaro sumability in my real analysis 2 course. Trying to pick up a bit of complex analysis on my own now though.

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Re: Bad math

Postby mike-l » Sun May 25, 2014 5:38 pm UTC

Yeah, complex analysis was only required for honors degrees. But nonetheless it was a third year course for everyone. There was a fourth year cross listed course that went into much more depth in both real and complex analysis (it used papa Rudin)
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Re: Bad math

Postby PM 2Ring » Mon May 26, 2014 11:04 am UTC

Eebster the Great wrote:It depends on the depth of the course. Some "complex analysis" courses are little more than basic calculus in the context of the complex derivative. Those could probably be taught in high school (though I don't think they are). Some are proof-based and start from the ground up. Those could technically be taught at any level, but tend to be difficult enough that underclassmen would usually be ill-advised to take them. And some are very high level and won't make any sense without substantial math experience to back it up. Those are mainly graduate level.

Here, math and engineering majors take complex analysis as undergrads, but hardly anyone else does.


Yeah, by "some exposure" I just meant at least having had a basic introduction to calculus in the context of the complex derivative. If you've studied complex numbers and some partial differentiation, it's not a huge leap to the Cauchy–Riemann equations and an introduction to the concept of holomorphic functions (with perhaps a mention of Riemann surfaces). With that background, it ought to be possible to have some fun with the gamma function.

OTOH, you don't even need all that stuff to be introduced to the gamma function with a real argument as an extension of the (shifted) factorial function. It's not that hard to show that Γ(n) = (n-1)! from the integral definition; that could possibly be done at the high-school level (but I guess it generally isn't).

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Re: Bad math

Postby Tirian » Tue May 27, 2014 1:05 am UTC

PM 2Ring wrote:OTOH, you don't even need all that stuff to be introduced to the gamma function with a real argument as an extension of the (shifted) factorial function. It's not that hard to show that Γ(n) = (n-1)! from the integral definition; that could possibly be done at the high-school level (but I guess it generally isn't).


At least in the US, probably not. Integration by parts is not tested on the AB calculus exam, so that lets out most students who take calculus in high school. For the students taking BC, it is arguably possible to do a think-aloud on "let's construct a continuous function on the strictly positive reals such that f(x)=(x-1)!", and they could get a review about integration by parts, FTC, and an exposure to the formal theory of recursive sequences. The class is such a death march that I wouldn't be surprised if teachers wouldn't be able to squeeze it into the school year, but I agree that sufficiently gifted and accelerated students would probably be able to follow the argument if they did some independent study.

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Re: Bad math

Postby Eebster the Great » Tue May 27, 2014 5:12 am UTC

We looked at the real branch of the gamma function in BC Calculus. Granted, our class was pretty accelerated relative to the normal AP curriculum, but I never found it particularly challenging myself.

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Re: Bad math

Postby Xami » Tue Jun 17, 2014 10:29 pm UTC

I too have a BA in math and have also found it left me just shy of some really cool sounding things that I was never exposed to. Earlier someone mentioned knowing just enough math to not be interesting and feel it perfectly describes my position. Luckily, I paired it with a computer science degree :)

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Re: Bad math

Postby mike-l » Wed Jun 18, 2014 5:13 am UTC

Xami wrote:Earlier someone mentioned knowing just enough math to not be interesting and feel it perfectly describes my position.


That was me, and it wasn't in reference to any particular level of math knowledge, instead it was in reference to people who feel like the knowledge they have is the end of the road. It was specifically in reference to "so the correct answer is to take the average of the two? That logic looks like its from a 3-year old. Math is way more beautiful than that kind of logic".

There's absolutely no shame in not knowing things, and indeed, even the best mathematicians will have areas of math where they know relatively little (though I'd be surprised if there was a single mathematical fact I know that, say, Terry Tao didn't). But a huge part of mathematics is generalization and finding ways to make things make sense where they don't under simpler definitions, starting from negative numbers making sense of subtraction when you want to subtract a bigger number from a smaller number, all the way up to 'crazy' things like orbifolds making sense where a manifold wouldn't (yes there are much crazier things, but I think that's a suitably advanced example), and the 1+2+3+...=-1/12 is a wonderful example of this that leads to all sorts of truly beautiful mathematics.

The people I was referring to were people who say 'this sum doesn't make sense in my first year calculus definitions, so anyone who tries to make sense of it is foolish and inelegant'. That ends the conversation before it has a chance to get interesting, which is why I said 'they know just enough to be wildly uninteresting to talk to'. On the other hand, someone who understands the ideas of convergence of series, but is willing to entertain the idea of more general methods of assigning values to series, even if they've never seen things like Caesaro summability or Zeta Regularizarion is exactly the kind of person who I want to have a conversation with.

Of course it's a difficult line to walk, if you go to far you'll spend your entire life entertaining cranks.
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Re: Bad math

Postby Eebster the Great » Wed Jun 18, 2014 5:54 am UTC

Orbifolds are "just" generalizations of manifolds. But they are definitely a graduate-level topic, so I agree they are pretty abstract and beyond the ken of the vast majority of non-mathematicians.

The idea of assigning sums to divergent series is a bit weird. Obviously it requires a different definition of "sum" than what is normally used, but that's OK because even just sums of convergent infinite series require a different definition than finite sums. The meaning of such sums is of course somewhat different from conventional infinite sums, since the definition is not the same, but I think if you explain to people that "sum" sometimes needs to be used in a more general sense in which it will not mean quite what they expect, they will realize that the results won't make sense in the context of finite summation.

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Re: Bad math

Postby mike-l » Wed Jun 18, 2014 6:28 am UTC

Eebster the Great wrote:Orbifolds are "just" generalizations of manifolds.

That was kind of my point. Negative numbers are 'just' generalizations of natural numbers so that subtraction always works, orbifolds are 'just' generalizations of manifolds so that taking the quotient by a group action works, and Riemann normalization is 'just' a generalization of summation via limits of partial sums so that it works in a broader context. Huge amounts of mathematics are done by 'just' generalizing things. If you'd like, an almost identical example to orbifolds that is somehow much more difficult is stacks as a generalization of schemes (I personally have a hate/hate relationship with stacks after years of trying and failing to grok them, despite them 'simply' being categorical orbifolds. On the other hand, I may only know enough about orbifolds to THINK I understand them)
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Re: Bad math

Postby Xami » Wed Jun 18, 2014 2:34 pm UTC

So there was some explanation of the -1/12 thing that was just too far over my head. I found one that I could keep up with, but something bothered me.
So the geometric series for 1/(1-x)^2 = sum of n*x^(n-1) so for x = -1 you get 1-2+3-4+5... and now you know this equals 1/4
Then you do some math I actually understand and get that summing the positive integers gives you -3 times that amount and bam, -1/12
But I'm under the impression that the geometric series thing only works when |x| is < 1, and here it equals 1.

Was there something wrong with this explanation? Or, what sort of reading am I going to need to begin to even think about building the foundation

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Re: Bad math

Postby gmalivuk » Wed Jun 18, 2014 4:43 pm UTC

Xami wrote:But I'm under the impression that the geometric series thing only works when |x| is < 1, and here it equals 1.
Well yeah, some definitions of "works" are going to be stretched a bit any time you're arguing that the sum of all positive integers has some specific finite value. One of the ways to do this is to more or less declare by fiat that certain identities hold outside the range in which they were originally proven.
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Re: Bad math

Postby mike-l » Wed Jun 18, 2014 5:53 pm UTC

Basically what gmal said. As a comparison, consider multiplication. We define this as repeated addition, but that only works for natural numbers. However, we can look at how it behaves on natural numbers to find rules that work on a broader set, and use those rules instead to allow us to multiply ngatiges, fractions, irrationals, complex numbers etc.

The same thing is happening here. The initial definition doesn't work for x=-1, but the rule does work there, so we can expand our definition to include -1.

We need to be careful, because this may break other rules, so you may not be able to depend on the result when you want to (eg, multiplying by a natural number makes the number larger in magnitude, but that's not true for multiplication extended to fractional values) but as long as you're aware of this restriction, you can still explore where your new definitions lead.


Edit: so here's a question for those who know more about this than I do, how unique is this answer? Ie, if I have two familes fn and gn of analytic functions, such that fn(a) = gn(a) for all n, and define F to be the analytic continuation of sum fm, G the continuation of sum gn, then, assuming everything exists, must F(a) = G(a)?

Edit 2: so the above is false (it may depend on the path taken even for the same f and g), but are there conditions that make it true? Eg sum fn and sum gn converge on a set with a as a limit point?
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Re: Bad math

Postby firechicago » Wed Jun 18, 2014 8:41 pm UTC

Eebster the Great wrote:The idea of assigning sums to divergent series is a bit weird. Obviously it requires a different definition of "sum" than what is normally used, but that's OK because even just sums of convergent infinite series require a different definition than finite sums. The meaning of such sums is of course somewhat different from conventional infinite sums, since the definition is not the same, but I think if you explain to people that "sum" sometimes needs to be used in a more general sense in which it will not mean quite what they expect, they will realize that the results won't make sense in the context of finite summation.

I think that a big part of why people are happy to call both finite sums and infinite convergent summations "sums," but get annoyed when people do the same with Cesaro Summations, is that the sums for convergent summations play nicely with finite sums in a bunch of ways that Cesaro summations do not. For example if I have an infinite sum an = a1 + a2 + ... which converges to A, and I add some constant c to both sides, everything works out just fine. We can see that an + c = c+ a1 + a2 + ... which converges to A + c. And the same is true if we multiply by c. In some sense we're safe talking about an equaling A in the same way that 1 + 2 = 3. If A is a Cesaro summation, though, we're no longer safe assuming this and we can no longer manipulate the sequence as if it was equal to its sum.

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Re: Bad math

Postby gmalivuk » Wed Jun 18, 2014 8:51 pm UTC

firechicago wrote:
Eebster the Great wrote:The idea of assigning sums to divergent series is a bit weird. Obviously it requires a different definition of "sum" than what is normally used, but that's OK because even just sums of convergent infinite series require a different definition than finite sums. The meaning of such sums is of course somewhat different from conventional infinite sums, since the definition is not the same, but I think if you explain to people that "sum" sometimes needs to be used in a more general sense in which it will not mean quite what they expect, they will realize that the results won't make sense in the context of finite summation.

I think that a big part of why people are happy to call both finite sums and infinite convergent summations "sums," but get annoyed when people do the same with Cesaro Summations, is that the sums for convergent summations play nicely with finite sums in a bunch of ways that Cesaro summations do not. For example if I have an infinite sum an = a1 + a2 + ... which converges to A, and I add some constant c to both sides, everything works out just fine. We can see that an + c = c+ a1 + a2 + ... which converges to A + c. And the same is true if we multiply by c. In some sense we're safe talking about an equaling A in the same way that 1 + 2 = 3. If A is a Cesaro summation, though, we're no longer safe assuming this and we can no longer manipulate the sequence as if it was equal to its sum.
It might not be true in general, but the two examples you picked still work with Cesaro summation.
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Re: Bad math

Postby firechicago » Thu Jun 19, 2014 12:23 am UTC

gmalivuk wrote:It might not be true in general, but the two examples you picked still work with Cesaro summation.

And now I realize that I had slightly but crucially misunderstood the definition of Cesaro sums.

Also, really cool stack exchange post on Cesaro sums and where they are useful (look at the first answer): http://math.stackexchange.com/questions ... -intuition

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Re: Bad math

Postby Spambot5546 » Thu Jun 19, 2014 2:54 pm UTC

Infinite series sums were a gotcha for me for a while until I asked an instructor why the Riemann-Zeta function came out to (pi^2)/6 when all of the terms being added were rational. She pointed out that (pi^2)/6 wasn't the sum, because it's an infinite series and therefore can't have an actual sum, it's just the limit of the sum.

I assume some similar nuance of terminology exists with these Cesaro sums.

Interestingly, to me at least, after reading "Cesaro sum" 20 or so times over the course of this thread I finally figured out why it sounds familiar. It's a song. Well, an interlude between songs, as Tool loves to include in their albums.
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Re: Bad math

Postby Eebster the Great » Thu Jun 19, 2014 4:16 pm UTC

Well it's the limit of partial sums, which by definition is the sum. The Cesaro sum is not necessarily equal to the limit of partial sums, because it is usually used to sum divergent series for which that limit does not exist.

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Re: Bad math

Postby mike-l » Thu Jun 19, 2014 6:56 pm UTC

That really just depends on your definition of sum. The simplest definition of sum being the addition of two numbers, in which case even finite series are not 'sums', but it's trivial to expand the definition to include finite repetition. You can stop there, or you can expand the definition to infinitely many terms whose partial sums converge. Again you can stop there, or you can expand again to some other methods of dealing with divergent series. You can even throw out some earlier definitions so that your concept makes sense in a broader sense but sometimes disagrees on previous values, although at that point it becomes more important to be explicit about what you mean. Before that point, its not a huge deal communication wise to be vague, as anything that makes sense to both parties will have the same values.

Saying an infinite series is not really a sum (because sums must be finite) is not unlike saying that -1 or 1/2 are not really numbers because numbers must be positive whole numbers
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Re: Bad math

Postby skullturf » Mon Jun 23, 2014 4:11 pm UTC

Spambot5546 wrote:Infinite series sums were a gotcha for me for a while until I asked an instructor why the Riemann-Zeta function came out to (pi^2)/6 when all of the terms being added were rational.


It's not really that weird for a sum of rational numbers to be equal to an irrational number -- as long as it's an infinite sum.

Consider a number known to be irrational, such as the square root of 2. Its decimal representation starts like

1.414213562373095

which certainly can be written as an infinite sum of rational numbers:

1 + 4/10 + 1/100 + 4/1000 + 2/10000 + ...

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Re: Bad math

Postby mike-l » Mon Jun 23, 2014 6:16 pm UTC

In fact, one way of definining the real numbers is pretty much 'include all convergent sums of rational numbers' (well technically sequences, but since the difference of two rationals is again rational this is equivalent)

In particular, I pretty much always think of a decimal expansion as an infinite sums of powers of 10.
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Re: Bad math

Postby Eebster the Great » Mon Jun 23, 2014 8:29 pm UTC

But I think it is pretty strange that the sum of positive integers can be a negative fraction.

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Re: Bad math

Postby Spambot5546 » Tue Jun 24, 2014 12:46 pm UTC

That's because it's not. Not in the sense that we normally mean "sum" where "the sum of 3 and 5 is eight" or even in the "the sum from n equals 0 to infinity of one over n factorial is e" sense where what you're actually talking about is the limit of the sum of an infinite series.

No, this is some other sum. If the second or third post in this thread is to be believed it's an "analytic continuation of the zeta function". My knowledge of the zeta function is pretty superficial, so why it makes it work that way is lost on me. At the danger of being completely wrong for the second or third time in a way that I'm sure would be embarrassing if I were capable of feeling that emotion I think I can sum up* this thread as "it's not a mind blowing revelation that addition doesn't work the way you think it does, it's just confusing terminology."

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Re: Bad math

Postby gmalivuk » Tue Jun 24, 2014 12:47 pm UTC

It's not confusing terminology, though, any more than it's confusing to say the square root of negative 1 is i.
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Re: Bad math

Postby Spambot5546 » Tue Jun 24, 2014 12:59 pm UTC

It confused me enough to start this thread. i^2 = -1 didn't motivate me to start any threads.
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Re: Bad math

Postby Eebster the Great » Tue Jun 24, 2014 4:04 pm UTC

Spambot5546 wrote:That's because it's not. Not in the sense that we normally mean "sum" where "the sum of 3 and 5 is eight" or even in the "the sum from n equals 0 to infinity of one over n factorial is e" sense where what you're actually talking about is the limit of the sum of an infinite series.

It is though. It's an extension of that idea. Specifically, if a series is summable by more "ordinary" means (e.g. finite summation, limit of partial sums), you will reach the same sum by zeta regularization. Only when those other means fail to give a result will there be a difference. This is similar to using real exponentiation to give results when integer exponentiation fails. x^3 is easy to understand as x*x*x, but x^pi cannot be defined in those terms. Instead, it is defined based on the analytical continuation. The main difference is that in the case of summation, it isn't obvious which function you should analytically continue to give results. There are actually multiple ways of doing it, but all of them should give the same answer.

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Re: Bad math

Postby mike-l » Tue Jun 24, 2014 4:42 pm UTC

A small quibble, x^pi can be defined by continuity, which is admittedly less confusing than analytic continuation. x^pi lies between x^3 and x^4, between x^3.1 and x^3.2 (x^3.1 is the positive number whose 10th power is x^31) etc, so the values are more or less 'where they should be'. x^i might be a better example (though it's infinite valued, which is confusing in its own right)

I do agree with the broader point though, and am happy calling -1/12 the sum of 1+2+3+...

--------------

There are actually multiple ways of doing it, but all of them should give the same answer.

I asked earlier for conditions which make that true. I'm not sure if they are known, but it's not true in general. For example if you wanted to sum 2^n/n and did so by analytically continuing x^n/n to x=2, you could get any odd multiple of pi*i (since we started with -ln x, which has infinitely many branches
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Re: Bad math

Postby Eebster the Great » Tue Jun 24, 2014 4:49 pm UTC

Well it's just a special case of analytical continuation in which only one value is reasonable for each real number (unlike tetration, where there are many reasonable continuations).

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Re: Bad math

Postby mike-l » Tue Jun 24, 2014 4:55 pm UTC

I didn't mean to imply that it wasn't analytic continuation, just that you didn't need it and so it seems less mysterious. Also see my edit, I'm still wondering about that question
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Re: Bad math

Postby Spambot5546 » Tue Jun 24, 2014 7:42 pm UTC

Eebster the Great wrote:
Spambot5546 wrote:That's because it's not. Not in the sense that we normally mean "sum" where "the sum of 3 and 5 is eight" or even in the "the sum from n equals 0 to infinity of one over n factorial is e" sense where what you're actually talking about is the limit of the sum of an infinite series.

It is though. It's an extension of that idea. Specifically, if a series is summable by more "ordinary" means (e.g. finite summation, limit of partial sums), you will reach the same sum by zeta regularization. Only when those other means fail to give a result will there be a difference. This is similar to using real exponentiation to give results when integer exponentiation fails. x^3 is easy to understand as x*x*x, but x^pi cannot be defined in those terms. Instead, it is defined based on the analytical continuation. The main difference is that in the case of summation, it isn't obvious which function you should analytically continue to give results. There are actually multiple ways of doing it, but all of them should give the same answer.

Are you referring to convergence tests? Because they're all pretty internally consistent, just not with zeta regularization. I checked the sum of the naturals against three or four of them before posting this, it diverged every time.

Which isn't inconsistent with what you're saying, of course. Applying the zeta function to a convergent series produces the limit of the series, and sometimes it works on divergent series so we say the result of the zeta function for that series is its sum. I've yet to get it to work (admittedly, I haven't been trying very hard), but wikipedia says it's true and not about to argue with that.

But saying "we can extrapolate from this and apply a finite value to that divergent series" isn't the same as "this series actually adds up to that". Any convergence test and even the definition of the limit of a series say it doesn't. Common sense does, too, but common sense has no place in mathematics. :-P
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Re: Bad math

Postby Eebster the Great » Tue Jun 24, 2014 8:05 pm UTC

Who says divergent series can't have a sum? All it means is that that limit does not exist.

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Re: Bad math

Postby gmalivuk » Wed Jun 25, 2014 4:32 am UTC

Spambot5546 wrote:But saying "we can extrapolate from this and apply a finite value to that divergent series" isn't the same as "this series actually adds up to that". Any convergence test and even the definition of the limit of a series say it doesn't. Common sense does, too, but common sense has no place in mathematics.
The only reason you think common sense applies in the case of convergent series is because you've had enough exposure to standard definitions that it feels natural to you. But just look at the vehemence with which people claim 0.9999... < 1 to see how uncommon that sense really is.

The fact is, you have to define what positional notation and infinite series even mean before you can state that 0.9999... = 1, and those definitions are not intrinsically any more "real" than the definitions of other types of sums for infinite series whose partial sums don't have a limit.
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Re: Bad math

Postby cyanyoshi » Wed Jun 25, 2014 5:18 am UTC

On a similar note, my intuition has been trained to tell me stuff like how ...999 (an infinite string of nines in base ten) equals -1 because if you add 1 to ...999, then you get ...000 which is of course zero. Alternatively, let ...999 = x. It follows that 10*x = x - 9, which can be solved to yield x = -1. This type of logic is indeed part of the framework for p-adic notation, and is analogous to two's compliment representation of negative numbers by computers.

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Re: Bad math

Postby PM 2Ring » Wed Jun 25, 2014 5:46 am UTC

I've been waiting for someone to mention p-adics since my post on divergent geometric progressions back on page 1. :)

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Re: Bad math

Postby Eebster the Great » Wed Jun 25, 2014 6:15 am UTC

P-adic numbers are weird as fuck. Where can I learn about them? Would they be discussed in an introductory course to number theory? I suspect not, but is there anywhere else I could appreciate them (to more than a "Wikipedia education") without a full pure math curriculum?


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