# Questions tagged [divergent-series]

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### Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...
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### A suggestion for a superlimit

I have a question about summation methods. A value is assigned to a divergent sum. All methods agree that $\texttt{super-}\sum_{k=1}^{\infty} k^p = -\frac{B_{p+1}^{+}}{p+1}$ where $B_{p}$ are ...
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### Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here) ...
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### Why we cannot speak about the main or natural regularization?

Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it". But in practice if we consider some ...
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### What is the regularized sum of the following series (sum of all primes but spaced with zeros in place of non-primes)?

The sum over primes: $$\sum_{k=0}^\infty \{\text{k if k is prime, 0 otherwise}\}$$ I know that there is no known method to ascribe a reasonable value to the sum of the primes https://www.quora.com/...
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### Were there attempts to express derivatives of Delta function as polynomials of Delta function?

Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory? Particularly, it could ...
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### Hausdorff methods of summation

From the book of Boss "Classical and modern methods in summability": "The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
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### What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]

I tried Mathematica, various regularization methods, including Borel, with no result. On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at ...
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### Regularization on divergent series [closed]

I want to compute $$\sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}.$$ This series is surely not convergent for any ...
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### Is this a Borel summable $S = \sum_{k=0}^\infty (-1)^k (k!)a_k$ with $a_k$ alternating sequence?

let $S = \sum_{k=0}^\infty (-1)^k (k!)a_k$ a divergent series such that $b_k=(-1)^k (k!)a_k >0$ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
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### What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
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### Why is it customary to have formal series at infinity in the context of resurgence and 1-summability?

In the context of the theory of 1-summability and resurgence, it is customary to deal with formal series "at infinity" rather than at $0$ $$\sum_{n=0}^{\infty}a_nz^{-n}$$ This is stated for example ...
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### Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parellel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...
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### Regularized sums of Mobius sequence

Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$? Experimentally this seems plausible (up through ...
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### What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?

Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider ...
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### Dimensional regularization in odd dimensions

I am not quite sure that my question below is appropriate for this site, probably it should be addressed to the physical commutity. But I hope that some (mathematical) physicists do attend MO. I have ...
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### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...
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### What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question) I think I understood the concept of fractional derivatives applied to ...
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### Asymptotic series

I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not ...
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### Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s :$ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
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### Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m: $$L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots$$ where I want to make ...
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### Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x$ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far] For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...
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### Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$

UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text. This is a concise version of this math.SE question of mine. I've got an answer ...
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### Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?

(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.) The Ramanujan ...