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!

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    $\begingroup$ @HandeBruijn interesting sure but I don't see much summability theory in it $\endgroup$ – Renato Faraone Sep 26 '17 at 13:17
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    $\begingroup$ see the similar question mathoverflow.net/q/3204/7710 $\endgroup$ – Gottfried Helms Sep 26 '17 at 19:29
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    $\begingroup$ @GottfriedHelms I'm always amazed by how summability has developed in over a century! The Devil's invention is so fascinating. $\endgroup$ – Renato Faraone Sep 26 '17 at 19:56
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    $\begingroup$ 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 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" $\endgroup$ – Gottfried Helms Sep 26 '17 at 20:17
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    $\begingroup$ @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) $\endgroup$ – Renato Faraone Sep 29 '17 at 22:10

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

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