Questions tagged [divergent-series]
The divergent-series tag has no usage guidance.
123
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Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
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2
votes
2
answers
185
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Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
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7
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1
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If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
- 1,550
6
votes
0
answers
158
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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 ...
- 1,550
4
votes
0
answers
126
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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 ...
- 1,550
4
votes
1
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$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$
I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
- 143
30
votes
3
answers
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Do roots of these polynomials approach the negative of the Euler-Mascheroni constant?
Let $p_n$ be the $n$th degree polynomial that sends $\frac{k(k-1)}{2}$ to $\frac{k(k+1)}{2}$ for $k=1,2,...,n+1$. E.g., $p_2(x) = (6+13x -x^2)/6$ is the unique quadratic polynomial $p(x)$ satisfying $...
- 19k
35
votes
2
answers
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1+2+3+4+… and −⅛
Is there some deeper meaning to the following derivation (or rather one-parameter family of derivations) associating the divergent series $1+2+3+4+…$ with the value $-\frac 1 8$ (as opposed to the ...
- 19k
-2
votes
1
answer
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Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 185
6
votes
1
answer
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Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
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4
votes
1
answer
201
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Does the analytical continuation of $\sum f(n) x^n $ always have a branch cut if $f(z)$ has a pole?
I suspect the answer to the title question is 'no', but I'm hoping to find an explicit counterexample. Also, I am requiring that $\sum f(n) x^n $ has a finite radius of convergence, otherwise, the ...
- 1,550
0
votes
1
answer
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A holomorphic function in the open unit disk satisfying certain properties
Does there exist a function which is holomorphic in $|z|<1,$ continuous in $|z|\leq1$ and such that the series $\sum |a_n|$ is divergent, where $a_n$'s coefficients in the Taylor series expansion ...
- 173
8
votes
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Switching the order of a summation and replacing a series by its analytical continuation
Background
A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
- 1,550
2
votes
0
answers
224
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Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 149
6
votes
0
answers
290
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Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
- 1,550
3
votes
0
answers
147
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The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question
I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
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1
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0
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Crazy conjecture about Bernoulli umbra and reference request
For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.
Yet, it still remains mistery what ...
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6
answers
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What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 1,550
2
votes
2
answers
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Assigning values to divergent oscillating integrals
I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...
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1
vote
1
answer
99
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On summation methods of divergent series
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\CC}{\mathcal C}$This previous question introduced the following notion of a summability space.
Let $\N:=\{1,2,\...
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3
votes
0
answers
325
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Extending reals with logarithm of zero: properties and reference request
If we take logarithmic function, we can see that its real part at zero approaches negative infinity with the same rate and sign from any direction on the complex plane, while the Cauchy main value of ...
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15
votes
2
answers
406
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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 ...
4
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1
answer
359
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How to correctly renormalize this function at the pole $x=1$? Evaluating: $\sum_{n=1}^{\infty} e^{e^n}$
So I was considering the divergent everywhere but 0 power series
$$ f(x) = \sum_{n=0}^{\infty} e^{e^n} x^n $$
Now one can do the following "questionable" manipulation
$$ f(x) = \sum_{n=0}^{\...
- 1,968
5
votes
3
answers
284
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Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 1,968
4
votes
2
answers
345
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Borel summation and the Abel function of $e^z-1$
This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
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6
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0
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286
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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1
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0
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Term-wise expectation of the Taylor series for $1/X$ yields asymptotic expansion for $\mathsf EX^{-1}$. What are the conditions?
Migrated from the MSE.
Let $X\sim F_X$ denote a continuous random variable. Computing the first negative moment $\mathsf EX^{-1}$ (assuming it exists) may not be tractable and thus a common tactic is ...
1
vote
2
answers
215
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What's the true regularized value of product of all natural numbers?
Muñoz Garcia and Pérez-Marco - The product over all primes is $4\pi^2$ claims that the regularized value of product $\prod_{k=1}^\infty k$ is $\sqrt{2\pi}$ and of $\prod_{k=1}^\infty p_k$ over primes $...
- 9,026
1
vote
1
answer
851
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Interchange summation order in the limit of number of elements going to $\infty$
Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
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1
vote
1
answer
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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^...
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4
votes
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 ...
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7
votes
2
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451
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Adrastus, Proclus, and 2+8+50+288+… vs. 1+9+49+289+…
According to the MacTutor essay "D'Arcy Thompson on Greek irrationals" (which I take to be a version of Thompson's original essay whose only liberty with the original text is giving English ...
- 19k
2
votes
1
answer
232
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Linear combinations of geometric series
Consider the uncountable-dimensional vector space $V$ consisting of finite linear combinations of infinite sequences of the form $(1,z,z^2,z^3,\dots)$ with $z \neq 1$ in $\mathbb{C}$. Since the ...
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2
votes
0
answers
90
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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
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0
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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 ...
- 1,550
3
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0
answers
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Arithmetic properties of error terms in divergent series
Most people know the famous equation $\sum_{k=1}^{\infty} k = -\frac{1}{12}$, justified for example by interpreting the LHS as $\zeta(-1)$.
My question: does the sequence $\{\frac{1}{12}+\sum_{k=1}^n ...
- 14.7k
3
votes
2
answers
385
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A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Introduction
I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
- 1,550
11
votes
1
answer
909
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Divergent series summation beyond natural boundaries
I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
- 1,550
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votes
2
answers
1k
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Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$
I'm hoping to find a reasonable value to assign to the divergent series $\sum_{n=0}^\infty (-1)^n n^n$ and $\sum_{n=0}^\infty (-1)^n (xn)^n$. For the first one, I have obtained something around 0.71, ...
- 1,550
2
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1
answer
212
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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 ...
- 1,550
4
votes
1
answer
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Is there a superpolynomial sequence which is Abel-summable?
A sequence $a_n \in \mathbb{C}, \ n = 1, 2, 3, \dots$ is Abel-summable if for all $|x| < 1$ the sum
$$g(x) = \sum_{n = 1}^{\infty} a_n x^n$$
converges and the limit $\lim_{x \to 1^{-}} g(x)$ exists....
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3
votes
0
answers
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Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
- 31
3
votes
1
answer
229
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Is there an asymptotic bound between converging and diverging series? [closed]
Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, ...
- 41
6
votes
0
answers
1k
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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 ...
- 1,550
1
vote
0
answers
91
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What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?
I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs.
So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
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2
votes
0
answers
211
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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 ...
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4
votes
0
answers
255
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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<...
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3
votes
0
answers
406
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What intuitive meaning "determinant" of a divergency (divergent integral or series) can have? [closed]
I am working on the algebra of "divergencies", that is, infinite integrals, series and germs.
So, I decided to construct something similar to determinant of a matrix of these entities.
$$\...
- 9,026
2
votes
0
answers
272
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Can be this "handwaving" idea about "counting" reals somehow put on solid ground?
We know that the Cantor's cardinality of a countable set is $\aleph_0$ and the cardinality of continuum is $2^{\aleph_0}=\aleph_0^{\aleph_0}$. Unfortunately, this measure is based on the idea of ...
- 9,026
10
votes
2
answers
894
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Converse to Erdős' conjecture on arithmetic progressions
I apologise in advance if this has been asked here before. I did a search and did not find anything obvious. Erdős' conjecture states that if $A\subseteq {\bf N}$ is such that $\sum_{n\in A} n^{-1}$ ...
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