Questions tagged [divergent-series]
The divergent-series tag has no usage guidance.
131 questions
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A family of divergent series
Is it reasonable to assert that $0^k - 1^k - 2^k + 3^k - 4^k + 5^k + 6^k - 7^k - ... = 0$ for all $k > 1$?
Here the signs are given by the Thue-Morse sequence; that is, the sign of $m^k$ is $+$ or ...
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Numerical algorithm for extracting the coefficients of transseries
Assume a function $f(x)$ is given numerically for $x>0$, i.e. for any $x>0$ there is a numerical procedure to obtain $f(x)$ to any desired precision.
Also assume that the function $f(x)$ has a ...
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Cesaro mean of iterates of function with non-attractive fixed point
Let $f : A \to A \subseteq \mathbb R$ be a real function with a fixed point $a_0 = f(a_0)$ which is not attractive.
Let $f^k = f \circ f \circ ... \circ f$ be the $k^{th}$ iterate of $f$ (with the ...
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On a particular case of Dirichlet series [closed]
I've this series:
$$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$
where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $.
I need the limit of series like an analytic function of $...
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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 n/\Phi}(p ...
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Can one define "Ramanujan Summation" over algebraic number fields?
With some trepidation, I ask to "evaluate" badly divergent sums. Generalizing $\sum n = -\tfrac{1}{12}$ what would be the value of this sum over $\mathbb{Z}[i]$?
$$\sum_{m,n \geq 0} (m+in) \hspace{0....
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Divergence of Dirichlet series
Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge?
I asked ...
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To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?
This is a cross-post from MSE.
In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees.
The idea to construct such a ...
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What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?
(I asked this in MSE before but there was only a general reference which did not help for my specific question)
I think I understood the concept of fractional derivatives applied to ...
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Is there an algebra for divergent series summation operators?
Let $D$ denote a divergent series and let $C$ denote a convergent series.
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
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Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?
Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
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Use of everywhere divergent generating functions
Generating functions are well-known to be much useful in combinatorics. But, maybe just since I am illiteral, all the applications coming in mind deal with power series, which are not just formal, but ...
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Regularized sums of Mobius sequence
Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$?
Experimentally this seems plausible (up through ...
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Summation mollifier to ensure a certain alternating series has the correct value
I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions:
(1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$
(2) $f(n,M) \...
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Is it possible to sum the divergent series with prime coefficients?
It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...
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Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,...$ (where $H(k)$ is the Hamming-weight)
In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...
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Is that series-transformation known in the context of divergent summation?
Note: I asked this question in math.stackexchange but did not receive an answer
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-...
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Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values
Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form
$S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$
where $H_k = \sum_{n=1}^{k} 1/...
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Efficient (divergent) summation for sum of zetas at negative arguments?
In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m:
$$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$
where I want to make ...
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Asymptotic series
I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not ...
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Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
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Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?
[This question is copied from math.stackexchange, it didn't get answers so far]
For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...
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Regularizing the divergent sum $1^k + 2^k + \cdots$
EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...
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Approximation:- Algorithmic considerations
Hello
I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...
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Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?
(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.)
The Ramanujan ...
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Closed form of divergent infinite product?
Okay, we know that
$$ \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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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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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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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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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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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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