
Terry Tao on how to compute non-converging infinite sums (2010) - ColinWright
http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/
======
drakaal
String theory likes this math because it assumes that there is a curve to the
sum, it will get smaller eventually. This helps make the "vibration" part of
sting theory work.

But just because you can prove something with math doesn't mean it is "real".

We all know that if you add any number of positive integers you get a positive
integer. This is very "provable".

The two are in contradiction. The sum of all natural numbers can't be 1/12th
if the sum of any two positive integers is another positive integer.

~~~
dtf
We all know that if you add any number of rational numbers you get a rational
number. Yet the sum of an infinite number of rational numbers can be
irrational (eg equation 2 in the article). Contradiction?

~~~
acjohnson55
Not sure that's any less intuitive than getting a rational number from an
infinite sum of integers, or a negative number from an infinite sum of
positives. It is indeed a paradox until you rigorously define what you mean by
"infinite sum", which is the whole point of the article :)

------
nate_martin
The non-converging sum 1 + 2 + 3 ... = -1/12 has applications in Bosonic
String Theory, interestingly enough
[http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%...](http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF)

~~~
adharmad
The non-converging sum you have mentioned is also called Ramanujan Summation.

------
mpyne
Wonderful article, though I got slightly distraught once he got through all
that hard math only to state essentially that it would get more interesting
below the fold.

I used to understand some of that (Taylor and Maclaurin series). I think the
"Integral Test" had been my high-water mark, it's amazing to see how much
further the mathematical concepts can be carried.

------
camperman
There's a wonderful proof of this on Numberphile's channel here:
[https://www.youtube.com/watch?v=w-I6XTVZXww](https://www.youtube.com/watch?v=w-I6XTVZXww)

~~~
jfarmer
Warning, prolix rant ahead.

I actually think that video is awful and it certainly doesn't illustrate what
Terry is illustrating in his blog post. Rather, the video goes through a bunch
of non-rigorous symbolic manipulations which ends with the author writing down
the sequence of symbols "1 + 2 + 3 + ... = -1/12".

However, unless one says precisely what one means by "+", "...", and "=" —
which Terry does — then we have no way of really saying whether the steps
taken to reach the so-called "conclusion" are valid or not. What's more, just
because the same sequence of symbols appear in both the video and Terry's blog
post doesn't mean they represent the same thing or have anything to do with
each other at all.

Put another way, if you and I reach the same conclusion, but you do so
rigorously and I do so speciously, that doesn't mean I've proven the same
thing as you have.

For example, in the video, why are we allowed to add together two infinite
series term by term in the way they describe? It seems "natural," I know, but
if that's natural, why can't we also, say, group the addition differently or
rearrange the terms? After all, a + (b + c) = (a + b) + c and a + b = b + a,
no matter what a and b are. Why can't we write

    
    
      S = 1 - 1 + 1 - 1 + ...
    

as

    
    
      S = (1 - 1) + (1 - 1) + ...
    

If we permit ourselves to do that, well, suddenly the sum "appears" to be 0
and not 1/2\. BTW, if you want a somewhat-more rigorous reason for why the sum
"should be" 1/2...

    
    
          S = 1 - 1 + 1 - 1 + ...
    
      So then
          
      1 - S = 1 - (1 - 1 + 1 - 1 + ...)
            = 1 - 1 + 1 - 1 + ...
            = S
    
      which implies S = 1/2
    

We're not proving that S = 1/2 here, though. We're proving this statement: if
it makes sense to talk about S and we're permitted to do the things we just
did to S then S = 1/2.

Terry knows all this, of course, which is why he says, "If one _formally_
applies (1) at these values of {s}..." That word "formally" is key here. To a
mathematician "formally" means "in a purely symbolic manner without
considering whether there's a sensible or consistent way of interpreting these
symbols."

So, Terry is saying, "If we treat these sums as purely symbolic entities then
when we substitute in s = -1 we get a the purely symbolic statement 1 + 2 + 3
+ ... = -1/12." He then goes on to illustrate ways we might make sense of this
purely symbolic (formal) sum.

The video, however, is no "proof" of anything at all. It's just a shell game
with symbols on a page, relying on people's vague intuition about what we're
allowed to do with numbers. Just because the symbols in the video include
those we typically take to represent numbers and addition doesn't mean they
actually do.

~~~
camperman
Thanks for the correction. I'm most definitely a layman when it comes to
infinite series. A couple of things gave me confidence in this video: the
result is presented in the string theory text and there's another video
demonstrating the same result using Riemann Zeta functions (so it must be
legit :)).

I sympathize with your frustration at the lack of rigor but isn't this kind of
like taking pot shots at a middle school physics textbook for not covering
Lagrangian mechanics?

~~~
jfarmer
No. While it's true that ζ(-1) = -1/12 and that the ζ function plays an
important role in physics, your reasoning is fallacious.

If X implies Y and we know that Y is true, that does not mean X is true. So
just because the video reached a "correct" conclusion does not mean that the
means by which they reached that that conclusion are sensible or even
consistent.

If I threw a dart at a dartboard labeled "What is 1 + 2 + 3 + ...?" and it
landed on the section marked "-1/12", would you believe my answer? Would the
fact that it happened to land on "-1/12" and also agreed with the ζ function
lend credibility to my dart-throwing method of proof?

Indeed, if I encapsulated the methods used in that video, I could use those
methods to have 1 + 2 + 3 + ... turn out to be any number I choose. This is
the problem with specious reasoning — one can use it to reach _any_
conclusion.

------
sillydonkey
I didn't read or listen to the OP, but 1+2+3+ ... = infinity, using infinity +
k = infinity, then k=0 for any k, then -1/3 = 0 = Whatever, then like Bertrand
Russel said, I am the Pope since 2=1 and the Pope and I are two people.

~~~
bren2013
[https://news.ycombinator.com/item?id=7083272](https://news.ycombinator.com/item?id=7083272)

------
gjmulhol
The first time I read this title, I read Terry Tate
([http://www.youtube.com/watch?v=RzToNo7A-94](http://www.youtube.com/watch?v=RzToNo7A-94)).
I thought "woah, triple-T is really changing careers!"

