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Tagged with ca.classical-analysis-and-odes divergent-series
10 questions
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votes
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Singularities at the circle of convergence: generalization of Cauchy-Hadamard theorem
Consider a series $\sum a_n z^n$ with finite radius of convergence $R$. Cauchy-Hadamard theorem gives $1/R = lim\ sup |a_n|^{1/n}$.
Q: Suppose for some reason (e.g. numerical) we know that there is ...
- 96
21
votes
1
answer
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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
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1
answer
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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
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If the generating function summation and zeta regularized sum of a divergent series exist, do they always coincide?
One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...
- 730
8
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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 zeta-...
- 5,342
9
votes
2
answers
2k
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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 ...
CommunityBot
- 1
13
votes
1
answer
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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 ...
CommunityBot
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8
votes
2
answers
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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
2
votes
3
answers
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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 ...
- 331
27
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4
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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^{+}}
> \...
CommunityBot
- 1