Questions tagged [divergent-series]

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A problem about the series $\sin(n^p)$ [closed]

Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$ is divergent
adobereader's user avatar
3 votes
0 answers
102 views

How are distributions and divergent series summations related?

When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients $$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...
Caleb Briggs's user avatar
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18 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 $\...
not all wrong's user avatar
4 votes
1 answer
370 views

Series convergence if $\sum a_n^2 < \infty$

There are quite a few simple results about convergent/divergent series derived from similar ones. Here is a question in the same spirit that I saw posted on another forum. Unfortunately, I don't have ...
Ivan's user avatar
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5 votes
1 answer
131 views

History of asymptotic expansion of Laplace’s method between Laplace and Erdélyi

In 1774 Laplace understood that $I≔∫\textrm{d}x \exp kf(x)$ for $k≫0$ can be estimated if one knows 2-jet of $f$ at its point of maximum (as $I₀ ≔ ∫\textrm{d}x \exp kf₀(x)$ with $f₀$ quadratic with ...
Ilya Zakharevich's user avatar
23 votes
1 answer
1k views

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 ...
Sidharth Ghoshal's user avatar
2 votes
2 answers
215 views

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_{...
Caleb Briggs's user avatar
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7 votes
1 answer
302 views

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 $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
Caleb Briggs's user avatar
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6 votes
0 answers
167 views

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 ...
Caleb Briggs's user avatar
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4 votes
0 answers
140 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 ...
Caleb Briggs's user avatar
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4 votes
1 answer
466 views

$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$

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 ...
Wreior's user avatar
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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 $...
James Propp's user avatar
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37 votes
2 answers
4k views

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 ...
James Propp's user avatar
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-2 votes
1 answer
203 views

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(...
MrPie 's user avatar
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6 votes
1 answer
224 views

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_{...
Caleb Briggs's user avatar
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4 votes
1 answer
233 views

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 ...
Caleb Briggs's user avatar
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0 votes
1 answer
162 views

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 ...
Nik's user avatar
  • 165
8 votes
0 answers
241 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. ...
Caleb Briggs's user avatar
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2 votes
0 answers
228 views

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 ...
Zaza's user avatar
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7 votes
0 answers
297 views

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 ...
Caleb Briggs's user avatar
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3 votes
0 answers
163 views

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^...
Caleb Briggs's user avatar
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1 vote
0 answers
100 views

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 ...
Anixx's user avatar
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19 votes
6 answers
1k views

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\...
Caleb Briggs's user avatar
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2 votes
2 answers
196 views

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 ...
Caleb Briggs's user avatar
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1 vote
1 answer
102 views

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,\...
Iosif Pinelis's user avatar
3 votes
0 answers
359 views

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 ...
Anixx's user avatar
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15 votes
2 answers
442 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 ...
Serge the Toaster's user avatar
4 votes
1 answer
378 views

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}^{\...
Sidharth Ghoshal's user avatar
5 votes
3 answers
320 views

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 ...
Sidharth Ghoshal's user avatar
4 votes
2 answers
387 views

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,...
Richard Diagram's user avatar
6 votes
0 answers
300 views

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 ...
Anixx's user avatar
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1 vote
0 answers
96 views

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 ...
Aaron Hendrickson's user avatar
1 vote
2 answers
236 views

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 $...
Anixx's user avatar
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1 vote
1 answer
2k views

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 ...
user1172131's user avatar
1 vote
1 answer
190 views

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^...
Caleb Briggs's user avatar
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4 votes
1 answer
163 views

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 ...
tmh's user avatar
  • 685
7 votes
2 answers
461 views

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 ...
James Propp's user avatar
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3 votes
1 answer
266 views

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 ...
James Propp's user avatar
  • 19.4k
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 $L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2$
Olivier Bodini's user avatar
3 votes
0 answers
221 views

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 ...
Caleb Briggs's user avatar
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3 votes
0 answers
146 views

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 ...
David Corwin's user avatar
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3 votes
2 answers
424 views

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 ...
Caleb Briggs's user avatar
  • 1,662
12 votes
1 answer
973 views

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 ...
Caleb Briggs's user avatar
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10 votes
2 answers
2k views

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, ...
Caleb Briggs's user avatar
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2 votes
1 answer
234 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 ...
Caleb Briggs's user avatar
  • 1,662
4 votes
1 answer
184 views

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....
Random's user avatar
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3 votes
0 answers
61 views

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 ...
popa13's user avatar
  • 31
3 votes
1 answer
254 views

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, ...
Niv Sarig's user avatar
6 votes
0 answers
2k views

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 ...
Caleb Briggs's user avatar
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1 vote
0 answers
98 views

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 ...
Anixx's user avatar
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