# Questions tagged [divergent-series]

The divergent-series tag has no usage guidance.

85
questions

**4**

votes

**1**answer

109 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....

**-1**

votes

**0**answers

49 views

### A convergence theorem of K Weierstrass [closed]

The theorem I am referring to is the following one:
Let $\{f_{n}\}_{n \in \mathbb{N}}$ a sequence of holomorphic functions in a domain $D$ which converges compactly there to $f \colon D \to \mathbb{C}...

**3**

votes

**0**answers

34 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 ...

**3**

votes

**1**answer

180 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, ...

**6**

votes

**0**answers

257 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 ...

**0**

votes

**0**answers

43 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 ...

**2**

votes

**0**answers

143 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 ...

**4**

votes

**0**answers

194 views

### 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<...

**3**

votes

**0**answers

371 views

### 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.
$$\...

**2**

votes

**0**answers

230 views

### 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 ...

**10**

votes

**2**answers

838 views

### 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 $\sum_{n\in A} n^{-1}$ ...

**3**

votes

**1**answer

325 views

### Exchanging series and integrals

I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...

**2**

votes

**2**answers

167 views

### What is the growth rate of the sum of powers of distinct primes closest to a given a integer?

Let $n$ be a positive integer, and
$$2 = p_1 < p_2 < \dots < p_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p_j^{e_j}$ be the largest power of $...

**8**

votes

**3**answers

510 views

### Discrete entropy of the integer part of a random variable

Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete ...

**1**

vote

**1**answer

62 views

### 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, \ \ \ \ \...

**1**

vote

**0**answers

56 views

### Finding a variable P for which a sum converges [closed]

I need guidance in finding a variable P for which $ \sum _{n=4}^{\infty }\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p $ converges, or proof that there doesn't exist such P ...

**2**

votes

**0**answers

90 views

### Theta-function in the lower half-plane

Standard theta function
$$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$
has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...

**6**

votes

**3**answers

312 views

### Is regularization of infinite sums by analytic continuation unique?

There are ill-posed summations that we can assign values to, take for concreteness,
$$ S = \sum_{k=0}^\infty k $$
to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...

**8**

votes

**2**answers

615 views

### Regularizing the sum of all primes

In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes?
$$ \sum_{p \text{ prime}} p $$
Neither of these questions obtained a ...

**1**

vote

**1**answer

406 views

### What are the consequences if we could express tangent via logarithm in an algebraic system? [closed]

Working on an algebra of divergent integrals I came to the following relation:
If $\tau=\int_0^\infty dx$ then
$$\ln (\tau+a)=\int_{0}^\infty \psi'(x+1/2+a)dx$$
and this directly gives the following ...

**1**

vote

**2**answers

432 views

### Generalized limits

Cross-posted from Math SE.
The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. It asks the following question:
...

**1**

vote

**1**answer

91 views

### Partitions of $\mathbb N$ which generate only divergent series of reciprocals of elements of those sets [closed]

This question is a result of some thinking about $\mathbb N$, divergent series and partitions of sets.
Although elementary, I am not skilled enough to answer it at the present moment.
Is it ...

**5**

votes

**0**answers

866 views

### Guessing of $n$th prime from “super- regularized” product of primes

( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.)
We know "super-...

**18**

votes

**1**answer

1k views

### Divergent Series & Continued Fraction (from Gauss' Mathematical Diary)

I've asked that question before on History of Science and Mathematics but haven't received an answer
Does someone have a reference or further explanation on Gauß' entry from May 24, 1796 in his ...

**3**

votes

**1**answer

154 views

### A suggestion for a superlimit

I have a question about summation methods. A value is assigned to a divergent sum. All methods agree that $\texttt{super-}\sum_{k=1}^{\infty} k^p = -\frac{B_{p+1}^{+}}{p+1}$ where $B_{p}$ are ...

**0**

votes

**0**answers

88 views

### Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)

(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...

**1**

vote

**1**answer

211 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\...

**3**

votes

**0**answers

336 views

### New/useful method for summation of divergent series?

Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...

**1**

vote

**1**answer

175 views

### Why we cannot speak about the main or natural regularization?

Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it".
But in practice if we consider some ...

**2**

votes

**0**answers

172 views

### What is the regularized sum of the following series (sum of all primes but spaced with zeros in place of non-primes)?

The sum over primes:
$$\sum_{k=0}^\infty \{\text{k if k is prime, 0 otherwise}\}$$
I know that there is no known method to ascribe a reasonable value to the sum of the primes https://www.quora.com/...

**-1**

votes

**2**answers

564 views

### 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 ...

**0**

votes

**0**answers

79 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 ...

**2**

votes

**0**answers

99 views

### What is the generalized sum of the following series? $\sum _{x=1}^{\infty } \sqrt{s^2 x^2-1}$ [closed]

I tried Mathematica, various regularization methods, including Borel, with no result.
On Math.SE the question was attacked with claims that divergent series cannot have a sum, so I decided to ask at ...

**1**

vote

**0**answers

174 views

### Regularization on divergent series [closed]

I want to compute $$ \sum ^{\infty }_{n=1}\dfrac {\left( 2n+k-2\right) !\zeta \left( 2n\right) \left( -1\right) ^{n-1}}{\left( 2\pi \right) ^{2n}}. $$
This series is surely not convergent for any ...

**0**

votes

**1**answer

165 views

### Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...

**1**

vote

**1**answer

152 views

### What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...

**0**

votes

**1**answer

72 views

### Why is it customary to have formal series at infinity in the context of resurgence and 1-summability?

In the context of the theory of 1-summability and resurgence, it is customary to deal with formal series "at infinity" rather than at $0$
$$
\sum_{n=0}^{\infty}a_nz^{-n}
$$
This is stated for example ...

**6**

votes

**1**answer

219 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}...

**2**

votes

**0**answers

132 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 ...

**3**

votes

**2**answers

293 views

### A family of divergent series

Is it reasonable to assert that $0^k - 1^k - 2^k + 3^k - 4^k + 5^k + 6^k - 7^k - ... = 0$ for all $k > 1$?
Here the signs are given by the Thue-Morse sequence; that is, the sign of $m^k$ is $+$ or ...

**0**

votes

**2**answers

132 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 ...

**5**

votes

**0**answers

91 views

### On a particular case of Dirichlet series [closed]

I've this series:
$$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$
where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $.
I need the limit of series like an analytic function of $...

**6**

votes

**3**answers

333 views

### Summating divergent series arising from the application of the Euler-Maclaurin formula to power law functions with non-integer exponents

As the title says, I'm stuck trying to find an expression for $\sum_{n=a}^b n^{q}$, with q being a positive rational number but not an integer, which does not demand unfeasibly long computation times ...

**9**

votes

**1**answer

406 views

### Can one define “Ramanujan Summation” over algebraic number fields?

With some trepidation, I ask to "evaluate" badly divergent sums. Generalizing $\sum n = -\tfrac{1}{12}$ what would be the value of this sum over $\mathbb{Z}[i]$?
$$\sum_{m,n \geq 0} (m+in) \hspace{0....

**2**

votes

**1**answer

269 views

### Division methods for divergent continued fractions

I hadn't even noticed before entering the subject above the parallel with the title of an earlier question posted here titled Summation methods for divergent series. And it's just mutatis mutandis as ...

**10**

votes

**1**answer

577 views

### Regularized sums of Mobius sequence

Do $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n/s}$ and $\lim_{s \rightarrow \infty} \sum_{n \geq 1} \mu(n) e^{-n^2/s^2}$ both equal $-2$?
Experimentally this seems plausible (up through ...

**1**

vote

**2**answers

90 views

### Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions:
(1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$
(2) $f(n,M) \...

**7**

votes

**1**answer

665 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**11**

votes

**1**answer

567 views

### 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 ...

**9**

votes

**1**answer

496 views

### To what extent is it possible to generalise a natural bijection between trees and $7$-tuples of trees, suggested by divergent series?

This is a cross-post from MSE.
In the paper "Seven Trees In One" by Andreas Blass, a "very explicit" bijection is found between trees and 7-tuples of such trees.
The idea to construct such a ...