The divergent-series tag has no wiki summary.

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### Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values

Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form
$S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$
where $H_k = \sum_{n=1}^{k} ...

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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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### ζ(-n) and “powers” of Grandi's series

For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+...$
A value can also be assigned to the related series ${n+0 \choose n}+{n+1 ...

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### How to find the coefficients of a poor-converging series?

I have the series
$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$
and the boundary conditions
$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...

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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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### Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...

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### Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...

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### Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...

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### Approximation:- Algorithmic considerations

Hello
I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...

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### Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and
$$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$
See ...

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### Abel summation of the alternating series of primes?

Consider the ordinary generating function of the sequence of primes ($2+3x+5x^2+7x^3 + ...$); by the ratio test and the prime number theorem, its radius of convergence is $1$. Thus, we might well ask ...

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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 ...

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### Is that series-transformation known in the context of divergent summation?

Note: I asked this question in math.stackexchange but did not receive an answer
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular ...

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### mertens-function in the light of divergent summation - what summation method were best adapted

Just reading about the Mertens-function in the other thread
Mertens function I remember an earlier attempt to apply divergent summation
to the series which is constructed of the Moebius-function
at ...

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### Uses of Divergent Series and their summation-values in mathematics ?

This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see.
Dear MO-Community,
When I was trying to find closed-form representations for odd ...

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### Is there a “quantum” Riemann zeta function?

Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the ...

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### Closed form of divergent infinite product?

Okay, we know that
$$ \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big) $$ .
Is there some known (trigonometric(?)) function that is equal to the following infinite ...

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### Summing a divergent series and a constant combined

At least according to the answer to this question, $\zeta(1) = \gamma $ (once reqularized, of course).
Let me rephrase that by stating that:
$$ \sigma(\zeta(1)) = \gamma $$
Here, $\sigma(x)$ is the ...

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### Non-absolute convergence of series with asymtotically equal coefficients

The following seems to be a question related to standard calculus, but I am not quite sure
where to look for an answer.
Suppose $f,g:\mathbb{N} \to \mathbb{C}$ are such that the have the same ...

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### Testing for asymptotic series

I would like to know if there is a systematic way of writing down an asymptotic series representation of a function?
(like one can use Taylor series expansion for doing a power series).
Conversely ...

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### What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups?

The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be ...

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### Values of cusp forms at q = 1 ?

Take a cusp form $f$ and let $f(q) = q + a_2q^2 + q_3q^3 + \ldots$" denote its $q$-expansion (assume that the $a_k$ are integers, and that $f$ comes from an elliptic curve $E$). Of course the series ...

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### Do Abel summation and zeta summation always coincide?

This is a more focused version of Summation methods for divergent series.
Let $a_n$ be a sequence of real
numbers such that $\lim_{x \to 1^{-}}
> \sum a_n x^n$ and $\lim_{s \to 0^{+}}
> ...

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### Summation methods for divergent series

There are many methods for assigning a value to a series that diverges, e.g. zeta function regularization, Abel summation, Cesaro summation, etc. From all of the examples I've found, two methods ...

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### Does any method of summing divergent series work on the harmonic series?

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic ...