All Questions
Tagged with divergent-series sequences-and-series
16 questions with no upvoted or accepted answers
16
votes
0
answers
1k
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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 ...
user11000
9
votes
0
answers
313
views
Switching the order of a summation and replacing a series by its analytical continuation
Background
A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
- 1,730
7
votes
0
answers
306
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Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
- 1,730
6
votes
0
answers
171
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Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
- 1,730
6
votes
0
answers
2k
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Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
- 1,730
4
votes
0
answers
160
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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
4
votes
0
answers
289
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Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$
Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function.
The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<...
- 203
3
votes
0
answers
171
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The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
- 1,730
3
votes
0
answers
276
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Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series
Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
- 1,730
3
votes
0
answers
79
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Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
- 31
2
votes
0
answers
90
views
Equality of bivariate formal series
Is it possible to prove algebraically that the two series uniquely defined by the following equations are equal:
$L_1=uz+zL_1^2+z \partial_uL_1$ and
$L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2$
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
2
votes
0
answers
143
views
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 ...
- 1,150
1
vote
1
answer
270
views
Calculus of variation with discontinuous solutions?
I'm thinking of the following question:
Consider a function $f: U\rightarrow\mathbb{R}$ where $U=[0,L_1)\cup(L_1,L]$, and an energy functional $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
- 9
0
votes
1
answer
271
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Solving (or approximating) a certain delay differential equation
I'm interested in finding the (unique?) solution to the set of delay differential equations
$$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$
$$f_x(w,x) = wf(w,w^2x)$$
With the initial condition $f(1,x) = e^...
- 1,730
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