Mark Chu-Carroll comments on this phenomenon here: http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/
I’m not familiar with series mathematics much, but his take on it is:
Comments from the more mathematically-inclined?
When one says 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + … = ln(2), that’s also just a way of assigning a value to the series. When one says 2 + 3 = 5, that’s also just a way of assigning a value to the series. There’s not some distinguished sense in which those accounts of summation describe the actual value those series are actually equal to, but this account of summation does not. That’s a false distinction. There is no “actual” value of a series, except relative to an account of summation. And there’s not a distinguished One True Account of summation.
(You could call one account of series summation the One True Account, of course, if you were particularly fond of it, and didn’t care about any others. Plenty of people do. But that is, literally, ignorant. It is ignoring all the other possibilities which are justified as capturing a sense of summation in just the same way as any other, by their analogies to other concepts in the web of things thought of as summation.)
I’m not sure I get it - so you’re saying there’s no difference between a Cesaro sum of a series and any other way of calculating a sum of a series?
There can be lots of differences. I’m saying there’s no good reason to say another kind of sum is the “actual” sum and the Cesaro sum is a spurious sum. There are just different kinds of sums; none of them is the “actual” one. Such talk is meaningless.
I think some of the flaws with MarkCC’s line of reasoning have already been pointed out.
Let’s be clear in the absolute strictest sense, under the normal definition of addition, you can only actually sum (that is add together each member of) a finite series. This is not quibbling, this is actually quite important. In order to get the sum of an infinite series you need to define what the sum of an infinite series is.
It was noticed a very long time ago that the limit of the set of partial sums allows you to recover the normal result for addition of finite series and also to define summation for some infinite series. After that it was noticed there were methods for extending summation to yet more infinite series and still being able to recover the results from previous methods.
What MarkCC seems to have missed is that it is just a case of what definition you use for summation. Sure, for example, the blokes in the numberphile video were playing it a bit loose when they didn’t mention a different definition of summation was being used than than the one on page six of Introduction to Real Analysis by A. N. Other, but they weren’t necessarily wrong to call it summation.
He also object because the answer is -1/12. Someone (I think ultrafilter) has already pointed out that if you add together a set of rationals then you always get a rational number, but the sum of a convergent series of rationals can be irrationals. Similarly the sum of a finite series is order-independent, but the sum of an infinite (conditionally convergent) series can be order dependent. So even if you just examine convergent series you can do things you can’t do with addition.
Couple of bloggers I follow have chimed in…
The sum of all natural numbers is not -1/12 (PZ Myers)
When Infinity Is Actually a Small, Negative Fraction (Phil Plait)
As usual, PZ should stick to biology. His quote contains an obvious logical error that others here have already pointed out:
There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn’t matter that infinity is involved…
By that logic, 4 - 4/3 + 4/5 - 4/7 … couldn’t possibly converge to pi, since a finite number of rationals always sums to a rational, and pi is irrational.
My question for Indistinguishable or others:
Is there any sense in which the harmonic series can sum to a finite value? Most of the series presented so far diverge fairly rapidly, but the harmonic series certainly doesn’t, and I wonder if this somehow makes a difference.
I have another question for Indistinguishable, et al.
Is there any serious disagreement on the answer to this problem among those in the upper levels of the Math community? Would you expect that anyone that won a Fields Medal would agree that the answer is correct?
There’s your taxonomy of summation methods, starting with absolute convergence, and working your way up. Is there any reasonable definition of a hierarchy of summations where a higher-numbered summation would give a different value than a lower-numbered summation? (For a case like your #6, which follows from #3, it wouldn’t count if it and, say, #5 gave different values.)
If a particular proposed summation method gave a different value than absolute convergence for a case where both gave values, I think I’d be hard-pressed to accept it over absolute convergence. That’s not to say that you can’t go beyond AC, where it doesn’t give a value. But calling AC the One True Account doesn’t seem too far-fetched, as long as “AC doesn’t give a value” isn’t conflated with “There is no value”. And similar if you want to take a different numbered method as your “One True Account”.
Are there any hierarchies you (or anyone) are aware of, where a higher level disagrees with a lower level? (Giving different values, not just something like “A gives a value for summation x and not summation y, but B gives a value for y and not x”.)
Integers and real numbers both have one traditional summation each.
But with each addition, it gets closer and closer to pi.
On the other hand with the sum of natural numbers it keeps on getting further and further away from -1/12. i.e. that is different logic.
Chronos:
Or maybe the answer is that natural number summation is a subset of integer summation which is a subset of real number summation which is a summation of complex number summation, etc. (maybe)
It never gets closer to becoming irrational, though. A number is either rational or not.
Besides, “closer and closer” only matters if you’re claiming that a series converges on a particular number. No one is claiming that 1+2+3+… converges on -1/12–just that it has a sum of -1/12 (for some definition of “sum”).
Ramanujan summation evaluates the harmonic series as the Euler-Mascheroni constant
Indistinguishable could answer this one better than me, but this is entirely about semantics, there’s no important mathematical principle at stake. When mathematicians use terms in a technical concept they’re usually careful to make sure they are clearly defined so as a) to stop any confusion b) to stop this sort trivial semantic argument.
I may be being a bit snide but I read another article on the Good Math, Bad math blog on the Uncertainty and it was absolutely awful, full of basic errors. For example he stated that the uncertainty principle meant that position was a discrete variable, however the uncertainty principle only applies to continuous (non-discrete) variables (and position must be continuous, in quantum mechanics at least).
(deleted)
<deleted as quoted deleted post>
I disagree with your version of someone else’s “logic” that it can’t converge to pi. I think it is irrelevant whether it is rational or irrational… the main thing is that it keeps on getting closer to pi. It doesn’t make sense that it would keep on getting closer to pi and then when you approach infinity it would suddenly stop converging… is there any example that you know of where it converges then suddenly doesn’t?
Also it is about an infinite sum of rationals, not a finite sum. The finite sum involves it approaching an irrational number, not being exactly equal to an irrational number.
The point is that contrary to his statement, infinity makes all the difference in the world.
This is true:
*There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. *
This is also true:
There is no mechanism in rational numbers by which addition of another rational number can produce an irrational number.
This is false:
It doesn’t matter that infinity is involved
Infinite sums just behave differently than finite sums. And again, we’re talking about sums here, not convergence. Everyone agrees that these are divergent series. That doesn’t mean we can’t assign a value to the sum.
I realize that I misspoke earlier (gotta be careful here). I said:
By that logic, 4 - 4/3 + 4/5 - 4/7 … couldn’t possibly converge to pi, since a finite number of rationals always sums to a rational, and pi is irrational.
I should have said:
By that logic, 4 - 4/3 + 4/5 - 4/7 … couldn’t possibly sum to pi, since a finite number of rationals always sums to a rational, and pi is irrational.
You are correct that the fault in his logic is not about convergence, but rather sums.