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What happens when you use summability methods on the harmonic series?

I'm quite surprised I haven't been able to find anything on this anywhere, considering that the partial sums of the harmonic series grow at a logarithmic rate, while series whose partial sums grow quadratically are summable.

Ayesha
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  • I googled "harmonic series grow .... quadratically are summable." and found the following paper which seems to discuss your question: http://arxiv.org/pdf/1303.1856.pdf – Squirtle Jan 25 '14 at 01:39
  • Look at Ramanujan Summation. I think for the harmonic series it gives Euler's $\gamma$. – André Nicolas Jan 25 '14 at 01:43
  • On the wiki page for cesaro sums it's stated that if a series diverges to infinity so does its cesaro sum. http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation – coffeemath Jan 25 '14 at 01:43
  • @coffeemath I'm not talking about Cesaro summation, rather, about summability methods in general. – Ayesha Jan 25 '14 at 01:44
  • @Ayesha Yes I got that from answers below, that other methods were of interest to you. Guess there are lots of them! – coffeemath Jan 25 '14 at 01:53
  • Did you overlook or exclude the Cauchy Condensation Test. It works for proof of divergence. Also it follows as a corollary about the series 1/n^p converging iff p>1. I would recommend to anyone else with this problem having a look at the second chapter of Abbott's Understanding Analysis. – smokeypeat Sep 25 '15 at 15:38
  • See https://math.stackexchange.com/questions/20005/is-it-possible-to-use-regularization-methods-on-the-harmonic-series and https://mathoverflow.net/questions/3204/does-any-method-of-summing-divergent-series-work-on-the-harmonic-series. – user76284 Jul 19 '19 at 10:59

4 Answers4

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The Ramanujan summation of the reciprocal of the positive integers is equal to the Euler-Mascheroni constant. Scroll down to the bottom of the page until where it says $$\sum_{n\geq1}^\Re \frac{1}{n}=\gamma$$

  • Is Ramanujan summation any different from other summability methods? The article seems to imply that this is so. – Ayesha Jan 25 '14 at 02:00
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Harmonic series is essentially $\zeta(1)$, Riemann Zeta function evaluated at $1$. While the function has a pole at $1$, we can find its Cauchy principal value there:

$$\lim_{h\to0}\frac{\zeta(1+h)+\zeta(1-h)}2=\gamma$$

It turns out to be Euler-Mascheroni constant.

Anixx
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What happens when you use summability methods on the harmonic series?

Most of them (Cesaro, Euler, Noerlund, Borel) diverge. I don't know whether the Aitken process can be made to give something.

The condition which allows the Cesaro/Euler/Noerlund/Borel to assign a value to a divergent series is when it has terms with alternating signs, or complex numbers. If they are applied to divergent series with strictly positive terms they all go to infinity. One must try whether there are possibilities for functional relations between nonalternating and alternating series which can then allow to sum the alternating series instead of the nonalternating one and then to recalculate the result using that functional relation (as it is done with the geometric series via the rational expression as a fraction ${ 1 \over 1-q}$ (except the pole at $q=1$) or with the Zeta-series and the functional relation with the alternating Zeta series in the way L. Euler had introduced it and is later made by the methods of analytic continuation).

Gottfried Helms
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  • What is the Aitken process? – Ayesha Jan 25 '14 at 19:03
  • A nonlinear(?) method for summing. I do not know really much about it; I just heard of it when I was in discussion with some mathematician about another divergent summation problem. He just used it in the way I did with Euler/Cesaro and my own invention to refute or confirm my results. Here is it in the english wikipedia https://en.wikipedia.org/wiki/Aitken%27s_delta-squared_process – Gottfried Helms Jan 25 '14 at 20:59
  • Hmm, perhaps the reference to the "Shanks method" https://en.wikipedia.org/wiki/Shanks_transformation is the correct one; however as I've understood the two methods are related to each other or follow some common rationale, at least they are two nonlinear working methods. – Gottfried Helms Jan 25 '14 at 21:02
  • This is a link to an older discussion (in 2007) with that correspondent in sci.math.research. To see "arguing with Shanks-method" in action search&find the keyword "shanks" in the text; The correspondent used it to confirm/refute my matrix based summation-method for a very strong diverging (while alternating in signs) series of the type $1-10+10^{10}-10^{10^{10}} + ... - ... $ – Gottfried Helms Jan 25 '14 at 21:25
  • Upps - I missed the link in the previous comment: http://go.helms-net.de/math/tetdocs/IterationSeriesSummation_1.htm – Gottfried Helms Jan 25 '14 at 21:31
  • For the record, the Shanks Transformation of the harmonic series is $S(A_\infty)=2+\sum_{n=2}^\infty (\frac2{n+1}-\frac1n)$, which behaves much like the original harmonic series. – Glen O Mar 27 '17 at 10:10
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and how about generalized harmonic series ??

there are two possible results

$$ \sum_{n=0}^{\infty} \frac{1}{(n+a)} = -\Psi (a) $$

and $$ \sum_{n=0}^{\infty} \frac{1}{(n+a)} = -\Psi (a) +log(a) $$

if $ a=1 $ we recover the euler mascheroni constant

Jose Garcia
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