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Studying summability theory I've come across many summation methods however by now I know only two not very interesting method which re-sums the harmonic series $\sum_{n=0}^\infty \frac 1{n+1}$ : the null method (most trivial of all, sums every series to $0$) and a generalization of Ramanujan summation (more interesting but not regular so not a summation method in the standard sense) for which

$$\sum_{n=0}^\infty \frac 1{n+1}=0\:\:\:(N)$$

$$\sum_{n=0}^\infty \frac 1{n+1}=\gamma \:\:\:(R')$$

Where $\gamma$ is the Euler constant.

Basically what I'm asking is: are there any other more or less known or useful method which can re-sum the harmonic series to a finite value? And more deeply why is it so difficult to regularize such series?

Any example or link to a paper would be very useful to me!

Keshav Srinivasan
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AlienRem
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  • Related? Both the Harmonic series and the Euler constant pop up in : What exactly are the curves that are a best fit to the Harmonic Cantilever? . – Han de Bruijn Sep 26 '17 at 13:15
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    @HandeBruijn interesting sure but I don't see much summability theory in it – AlienRem Sep 26 '17 at 13:17
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    see the similar question https://mathoverflow.net/q/3204/7710 – Gottfried Helms Sep 26 '17 at 19:29
  • @GottfriedHelms I've read it and found very interesting the answer by Steven but I don't have the background to understand it, what should I study in order to fully understand those concept? – AlienRem Sep 26 '17 at 19:36
  • Urrrxxx - that answer of Steven is also way over my head... – Gottfried Helms Sep 26 '17 at 19:48
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    @GottfriedHelms I'm always amazed by how summability has developed in over a century! The Devil's invention is so fascinating. – AlienRem Sep 26 '17 at 19:56
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    Renato - I like like you that "divergent series" subject. I've played around with a self-made matrix summation procedure based on the eulerian-numbers. That summation employs basically the same series-transformation as the Borel-summation does. Unfortunately I don't arrive at something else than infinity. If you like to read explorative articles then you can look at http://go.helms-net.de/math/index.htm for the two links at the entry "A matrix-method for divergent summation using the matrix of Eulerian numbers" – Gottfried Helms Sep 26 '17 at 20:17
  • @GottfriedHelms I'm far from graduating but my thesis will surely be on divergent series so everything you send me is greatly appreciated! – AlienRem Sep 26 '17 at 20:21
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    Hmm, perhaps I should take some effort to formulate some of my heuristical approaches as answerable questions - I've experimented with some matrix-methods, generalizations mainly of the Euler-summation ; but it is a couple of years ago and I don't remember whether/how I could possibly focus the unproven heuristics such that I could ask it here... perhaps I can come back to this next days. – Gottfried Helms Sep 26 '17 at 20:48
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    It seems misleading to call this a "sum," in particular, why should the sum of an infinite number of positive terms be less than the sum over a finite number of them? Why not just call this a map from the set of real-valued sequences to the set of real numbers, i.e., $h:\mathbb{R}^{\mathbb{N}} \rightarrow \mathbb{R}$ with $h({1/n}_{n=1}^{\infty}) = \gamma$? And then of course you should define the map $h$ precisely. – Michael Sep 27 '17 at 16:23
  • @Michael I prefer to use the term "resummation" instead of "sum" but in the context of summability it is pretty clear what we mean by "sum" and well... it is shorter to write – AlienRem Sep 27 '17 at 16:31
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    @RenatoFaraone : I'm not sure what you mean by "it is pretty clear." Once you start reusing standard $\sum$ notation, then it is not clear when you mean the usual definition and when you mean some other definition. Kind of like using the symbol $\sqrt{2}$ to mean $100$. – Michael Sep 27 '17 at 17:11
  • @Michael well imagine you are doing modular aritmethic, the standard notation is due to Gauss (if I remember correctly) but in a 10 page long where I am doing calculations mod 11 if I use the '=' sign and nothing else I think no one will say "I don't understand this notation" . The same is for summability, when we talk about generalized sum/resummation and we only use the term sum I don't see much a big ambiguity – AlienRem Sep 27 '17 at 17:16
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    @RenatoFaraone : In that case I encourage you to define your sum at the start. I googled "A generalization of Ramanujan summation" but could not find a precise definition (and of course there can be many different generalizations...). – Michael Sep 27 '17 at 17:38
  • @Michael you can find an entire book by the Springer about Ramanujan summation and its different definitions – AlienRem Sep 27 '17 at 17:40
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    Related: Can we assign a value to the sum of the reciprocals of the natural numbers? – Simply Beautiful Art Sep 29 '17 at 22:02
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    @SimplyBeautifulArt thanks! I've rad similar questions on this and other sites however what I'm specifically looking for is some other more well behaved (regular, maybe stable) method which assigns a finite sum to the harmonic series and maybe it would be great to understand why more a series diverges slowly more it is hard to resum it (this may sound general but knowing a bit of summability it should be clear enough) – AlienRem Sep 29 '17 at 22:10
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    The null method is always lame =P – Simply Beautiful Art Sep 29 '17 at 22:53
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    @SimplyBeautifulArt yeah never surprising but incredibly strong and also linear! Not that bad ;) – AlienRem Sep 29 '17 at 22:55
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    X'D True. ${~}$ – Simply Beautiful Art Sep 29 '17 at 23:01
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    Does this answer your question? Is it possible to use regularization methods on the Harmonic Series? – tparker Dec 04 '21 at 17:19

1 Answers1

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As an exercise in learning about "divergent series" and their classical summation-procedures I played with a method which is based on a matrix-transformation using the "Eulerian numbers". It is often "stronger" than Euler-summation (of any order), can sum not only the geometric series with $q<1$ but also the alternating series of factorials (it is similar to Borel-summation) - but cannot sum the nonalternating versions of that series when they are classically divergent, and thus cannot sum not the harmonic series.

The pure series transformation using the matrix $E$ from a series $a_1+a_2+a_3+...$ into a vector $T$ containing the transforms looks like $$ [a_1,a_2,a_3,...] \cdot E = [t_1, t_2,t_3,...] = T $$ and for convergent and summable cases the partial sums of the $t_k$ approximate the sum of the series-terms on the lhs and we have then a usable series-summation method (it gives also convergence acceleration for the summable cases). For reference, I call this provisorically the "Eulerian transform" or "Eulerian summation".

For the given case of the harmonic series the method gives for the partial sums of the $t_k$ infinity and thus the harmonic series is not summable by the "Eulerian summation".

But besides this that partial sums give the following remarkable approximation (writing the $N$'th harmonic number as $H_N$) $$ \sum_{k=1}^N t_k - H_N = \log 2 +\varepsilon_N \qquad \qquad \lim_{N \to \infty} \varepsilon_N=0$$ which reminds of the formula for the Euler-Mascheroni-constant, but has a different constant term.

Remark: the method has not yet a sufficient theoretical framework but because the entries in $E$ have a tractable composition one can do relatively simple partial analyses for the self-study. See initial exploration of E and the basic idea of summation method and for the current problem the investigation of the transform for some non-summable cases
Gottfried Helms
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