# Ramanujan Summation and ways to sum ordinarily divergent series

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Mentor
I'd just like to confirm that if the summation over k goes to infinity (as it does in the above version) then we don't actually need R at all. So is it true that R is an error term that is non-zero only if we truncate the summation terms over k at some k = p?
Actually using a non-rigorous derivation the R doesn't even appear.

Note here the sum is from 0 to n-1.

To make the above rigorous, should that appeal, see:
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1155-7.pdf
But I personally would not worry. Arguments like that are used in physics and applied math all the time - if you get too caught up in it you will find it takes up your time for no gain in using it to solve problems. But sometimes you just can't resist - I now that feeling only too well.

Have fun.

Thanks
Bill

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PAllen
Just want to add this link, which covers Cesaro summation and analytic continuation (the latter, a bit simplified):

WWGD
Gold Member
Just want to add this link, which covers Cesaro summation and analytic continuation (the latter, a bit simplified):
Because it depends on what " is" is. It does not mean standard convergence. I know you know this but I think most of those who watch the video don't.

PAllen
Because it depends on what " is" is. It does not mean standard convergence. I know you know this but I think most of those who watch the video don't.
Well this video, unlike the numberphile video it debunks, is very clear on the distinctions between different types of summation. It actually does not introduce Ramanujan summation. Instead it covers analytic continuation, and the computation of the zeta function in terms of the eta function.

WWGD
Gold Member
Well this video, unlike the numberphile video it debunks, is very clear on the distinctions between different types of summation. It actually does not introduce Ramanujan summation. Instead it covers analytic continuation, and the computation of the zeta function in terms of the eta function.
Ah, my bad for being lazy and making unwarranted assumptions.

Mentor
Well this video, unlike the numberphile video it debunks, is very clear on the distinctions between different types of summation. It actually does not introduce Ramanujan summation. Instead it covers analytic continuation, and the computation of the zeta function in terms of the eta function.
Exactly. Its one of the better ones around that makes it clear its simply how we define infinite summation - and analytic continuation, going to the complex plane, is a very natural way of extending it. Is why nearly all the guff you find on it posted on the internet is wrong. An interesting exercise for the advanced that sheds further light on it, is its relation to the Hahn-Banach theorem. Just as a start on that journey:
http://oak.conncoll.edu/cnham/Slides6.pdf
The reason Ramanujan Summation works for summing divergent series is, as mentioned in the rather good Math-lodger video, analytic continuation. Taking the Zeta function as an example it is fine for s >1, the C Ramanujan defines as the Ramanjuan sum is the same as the usual sum. But for other values the sum is divergent in the usual sense, but C still exists, and by analytic continuation must be the same as other methods, providing it is analytic, which the Ramanujan sum is. The Hahn- Banach theorem approach provides another interesting way of looking at it.

Thanks
Bill

WWGD
Gold Member
Exactly. Its one of the better ones around that makes it clear its simply how we define infinite summation - and analytic continuation, going to the complex plane, is a very natural way of extending it. Is why nearly all the guff you find on it posted on the internet is wrong. An interesting exercise for the advanced that sheds further light on it, is its relation to the Hahn-Banach theorem. Just as a start on that journey:
http://oak.conncoll.edu/cnham/Slides6.pdf
The reason Ramanujan Summation works for summing divergent series is, as mentioned in the rather good Math-lodger video, analytic continuation. Taking the Zeta function as an example it is fine for s >1, the C Ramanujan defines as the Ramanjuan sum is the same as the usual sum. But for other values the sum is divergent in the usual sense, but C still exists, and by analytic continuation must be the same as other methods, providing it is analytic, which the Ramanujan sum is. The Hahn- Banach theorem approach provides another interesting way of looking at it.

Thanks
Bill
Thank you, will look into it but it seems a bit confusing in that Hahn Banach is used to extend linear/sublinear maps from subspaces into the "host" superspace but I dont see how this applies to Taylor series which are not linear.

S.G. Janssens
Thank you, will look into it but it seems a bit confusing in that Hahn Banach is used to extend linear/sublinear maps from subspaces into the "host" superspace but I dont see how this applies to Taylor series which are not linear.
I will not comment on Ramanujan Summation itself, but regarding Hahn Banach: In certain cases, the theorem can be used to extend multilinear forms as well. Given that the $n$th term of a Taylor series (of a function defined on an open subset of a Banach space) is an $n$-linear form (actually evaluated at $n$ times the same argument) I could imagine that Hahn-Banach has its uses there.

The reason Ramanujan Summation works for summing divergent series is, as mentioned in the rather good Math-lodger video, analytic continuation.
Perhaps you have seen this blog post by Terence Tao. If you haven't, you'll find it interesting because Tao's aim here is to derive some stuff independently of analytic continuation -- things that are formally considered accessible only by stepping off the real line and wandering around the complex plane.
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

To make the (use of symbolic manipulations of differential operator D) rigorous, should that appeal, see:
http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1155-7.pdf
In my engineering math courses, we had to get used to the idea that you could write polynomials in D, and evn transcendental functions of D like $e^{Dh}$. At that time our priority was to pass the exams and get on with life, so no one spared any time to wonder how this actually works.

Now that I'm retired, I can afford to spend some time down these rabbit holes, purely as a hobby. Unfortunately, this article seems to be quite a bit beyond my grasp because it demands a certain level of understanding of abstract math.

I'm wondering how feasible the following approach would be, at least as a for-dummies picture:
Although "D" is not a number, it does take a function f(x) and give you f'(x), so in a sense we can think of D as a sort of number that represents the local value of f'(x)/f(x). If we plug that local ratio into a power series, then $e^{Dh}$ sort of makes sense. How far can we get if we try to run with this ball?

Edit:
Oops, the first problem is that $D^2$ is not necessarily the square of f'(x)/f(x).

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Mentor
Although "D" is not a number, it does take a function f(x) and give you f'(x), so in a sense we can think of D as a sort of number that represents the local value of f'(x)/f(x). If we plug that local ratio into a power series, then $e^{Dh}$ sort of makes sense. How far can we get if we try to run with this ball?
That's fine intuitively. You hit on the exact difference between applied and pure (or is that puerile ) math. In applied math degrees you still do some pure math simply as background (you should anyway). For example in the degree I did we did linear algebra as pure math before we did another subject Applied Linear Algebra. They don't tend to do that in Engineering and Physics courses so it makes following advanced pure math papers hard. Its actually a problem in applied math type degrees nowadays as well. My old alma mater doesn't even do the basic background pure math subjects because students thought it was just mind games. IMHO it's a big issue.

If you want to understand it you need a course in analysis, (colloquially called doing your epsilonics) which my alama mata once did but no longer does. A good reference and well written path into it for those already mathematically advanced is:
http://matrixeditions.com/5thUnifiedApproach.html
If you just have been exposed to basic calculus, and not gone deeper into applied math, I would suggest the following first:
https://www.amazon.com/dp/0691125333/?tag=pfamazon01-20
Thanks
Bill

Mentor
Perhaps you have seen this blog post by Terence Tao.
Always read Terry's Blog. It is probably my favorite blog on the whole internet

My favorite way into this stuff is Borel Summation which is so simple I cant resist detailing a link to it here:
https://www.nbi.dk/~polesen/borel/node7.html
It can't be used to directly sum the Zeta function, but there is another function called the Eta function defined by η(s) = 1 - 1/2^s + 1/3^s - 1/4^s ........ that you can easily derive a simple relation to the zeta function:
https://proofwiki.org/wiki/Riemann_Zeta_Function_in_terms_of_Dirichlet_Eta_Function

The Eta function is Borel Summable so low and behold you have summed the Zeta function.

Its interesting to see where analytic continuation has been used. I will leave those silly enough to read my ramblings to think about it.

Thanks
Bill

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bhobba said:
I like Hardy and his 'chatty' style, but it's a bit dated.
I think that Hardy's '1729' story was not true. I think that Hardy was aware of Ramanujan's prior writings regarding that number when the 2 men met in Punjab. I firmly disbelieve Hardy's anecdote to the effect that he had remarked to Ramanujan that he (Hardy) on his way to the meeting had ridden in taxicab number 1729, and that he (Hardy) thought 1729 was a dull or boring number, and that Ramanujan had immediately replied that 1729 was the least number that could be expressed as the sum of 2 cubes in 2 different ways -- despite Hardy's self-effacement and praise of the great mathematician Ramanujan, I believe that he just plain made up that story.

A characteristic of 1729 that Hardy did not report Ramanujan to have stated is that in duodecimal it is 1001 which numeral sequence in binary logic may be used to represent the 'if and only if' relation.

I think that Hardy intentionally fabricated the 1729 Hardy-Ramanujan anecdote.

Let's take the series 1 + 2 + 3 + ....

Let's add to it another series, that looks like

which is based on a normal distribution centered around say 20, with a spread of around 4, and the sum of the terms is 1.

Firstly, is this sum of two functions a valid candidate for a summation attempt? If so, it should result in -1/12 + 1, right?

If we look at the Ramanujan sum, then the bump around 20 would not contribute much to the result, would it? We would have -1/12 from the first series, and then we would have contributions from various derivatives of the "bump" taken at zero. These derivatives are going to be pretty negligible so the final result would still be about -1/12.

So is this connected to why Hardy cautioned about using the Ramanujan sum?

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Mentor
So is this connected to why Hardy cautioned about using the Ramanujan sum?
See the Wikipedia article on Divergent Series:
https://en.wikipedia.org/wiki/Divergent_series#Ramanujan_summation

Note what it says about Ramanujan Summation:
The Ramanujan sum of a series f(0) + f(1) + ... depends not only on the values of f at integers, but also on values of the function f at non-integral points, so it is not really a summation method in the sense of this article.

That's its problem, although I do not know an example, you can get a different answer for the same sum by using a different f, so requires caution. In your example, since linearity does not apply to Ramanujan summation, you can't add the results like you want to.

Thanks
Bill

you can get a different answer for the same sum by using a different f
That's an interesting point that I've missed in my (unsystematic) reading and video-viewing so far.
One way to generate new functions passing through the same sequence would be to add in one or more sine waves that go through zero at every integer. I would imagine that the derivatives of the sine wave around zero would dominate that puny -1/12 and change everything. (Unless they canceled out in some weird way).
It seems that the Ramanujan sum is really just a property of f around zero, that happens to equal the series sum for certain well-behaved types of f.

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Mentor
It seems that the Ramanujan sum is really just a property of f around zero, that happens to equal the series sum for certain well-behaved types of f.
Could be. I have a copy of Hardy's book but have not studied it as thoroughly as I would like amongst all the myriad of other things I want to study. He likely has a more detailed analysis. It so bad I set myself the goal 10 years ago of studying Weinberg's masterpiece on QFT - but am lagging well behind - at the moment I am still stuck on Banks - Modern QFT after QFT for the Gifted Amateur. The sojourn I did with Zee didn't help. At first I rather liked Zee, but as time went by it was too disjointed and not that well explained for my taste, Banks I found a lot better.

Thanks
Bill