All Questions
Tagged with divergent-series reference-request
19 questions
21
votes
1
answer
1k
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Does summing divergent series using cutoff functions give consistent results?
One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function:
$$
S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right)
$$
where $\...
- 7,817
4
votes
0
answers
160
views
Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
1
vote
0
answers
116
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Crazy conjecture about Bernoulli umbra and reference request
For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.
Yet, it still remains mistery what ...
- 10.1k
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
- 128k
4
votes
1
answer
195
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Reference request: Rigorously solving ODEs using divergent asymptotic series
In my research I have come across a divergent asymptotic series $\sum_{n =0}^\infty a_n f_n(x)$ that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond ...
- 41.1k
2
votes
1
answer
247
views
List of assigned values of divergent series
I'm hoping to find a list of divergent sums where the assigned value is generally accepted. For instance $\sum_{n=0}^\infty (-1)^n$ is generally accepted to be $\frac{1}{2}$. Moreover, its agreed upon ...
- 10.1k
2
votes
0
answers
232
views
Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
- 10.1k
28
votes
5
answers
6k
views
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 ...
- 455
1
vote
1
answer
88
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Convergence properties of related series
Let $u_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g_n\}_{n \in \mathbb{N}}$, $g_n \to 0$, such that
$$\sum\limits_{n \in \mathbb{N} } g_n = \infty, \ \ \ \ \...
- 24.2k
-1
votes
2
answers
850
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Were there attempts to express derivatives of Delta function as polynomials of Delta function?
Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory?
Particularly, it could ...
- 10.1k
0
votes
0
answers
89
views
Hausdorff methods of summation
From the book of Boss "Classical and modern methods in summability":
"The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
- 109
12
votes
2
answers
2k
views
Dimensional regularization in odd dimensions
I am not quite sure that my question below is appropriate for this site, probably it should be addressed to the physical commutity. But I hope that some (mathematical) physicists do attend MO. I have ...
- 1,996
6
votes
1
answer
326
views
Riemann surface from Riccati equation
I have quite a practical question motivated by physics.
Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum:
$$
(p(x))^2 + \dfrac{\hbar}{i}...
- 91.8k
0
votes
2
answers
188
views
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 ...
- 49
4
votes
1
answer
448
views
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-...
- 5,342
4
votes
3
answers
377
views
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 ...
- 311
8
votes
2
answers
2k
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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 ...
CommunityBot
- 1
5
votes
3
answers
424
views
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 ...
- 57.6k
20
votes
2
answers
1k
views
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 ...
- 27.7k