The divergent-series tag has no usage guidance.

**3**

votes

**0**answers

78 views

### How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...

**5**

votes

**3**answers

180 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 ...

**7**

votes

**1**answer

348 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) ...

**1**

vote

**1**answer

103 views

### Division methods for divergent continued fractions

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

**9**

votes

**1**answer

464 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

77 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) ...

**3**

votes

**1**answer

320 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 ...

**8**

votes

**0**answers

220 views

### If the generating function summation and zeta regularized sum of a divergent 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 ...

**7**

votes

**1**answer

316 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 ...

**12**

votes

**1**answer

431 views

### 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 ...

**2**

votes

**1**answer

131 views

### Alternating series $\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$ and multiple zeta values

Motivated by analytic continuation of solutions of a Picard-Fuchs equation, we encountered sums of the following form
$S(z;p)=\sum_{k=1}^{\infty}(-1)^{k+1} (H_k)^p z^k$
where $H_k = \sum_{n=1}^{k} ...

**5**

votes

**1**answer

483 views

### What are some geometric / physical / probabilistic interpretations of the Riemann zeta function at integer arguments n ≤ 1?

Introduction: This is slightly edited and generalised version of a question I asked on the Physics Stack Exchange website. This question has a twin brother asked here on MO, only now we consider ...

**8**

votes

**1**answer

378 views

### ζ(-n) and “powers” of Grandi's series

For n a non-negative integer, $ζ(-n)$ can be interpreted as assigning a value to the (divergent) series $1^n+2^n+3^n+4^n+...$
A value can also be assigned to the related series ${n+0 \choose n}+{n+1 ...

**6**

votes

**1**answer

625 views

### How to find the coefficients of a poor-converging series?

I have the series
$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$
and the boundary conditions
$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...

**4**

votes

**5**answers

300 views

### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...

**3**

votes

**1**answer

476 views

### What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question)
I think I understood the concept of fractional derivatives applied to ...

**3**

votes

**3**answers

259 views

### Asymptotic series

I have found many references to Poincaré and Borel in relation to their work on asymptotic series, but so far, every source I can get my hands on is very old, hence hard to read (this is not ...

**10**

votes

**1**answer

574 views

### Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series.
Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...

**6**

votes

**1**answer

387 views

### Efficient (divergent) summation for sum of zetas at negative arguments?

In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter m:
$$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$
where I want to make ...

**0**

votes

**1**answer

203 views

### Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far]
For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...

**7**

votes

**2**answers

2k views

### 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 ...

**27**

votes

**3**answers

2k views

### Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = ...

**14**

votes

**0**answers

895 views

### 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 ...

**6**

votes

**2**answers

912 views

### 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 ...

**9**

votes

**2**answers

2k views

### Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...

**2**

votes

**1**answer

401 views

### Approximation:- Algorithmic considerations

Hello
I want to approximate a function $f$ on $(a,b)$. The function is singular at the points $a$ and $b$, however I have asymptotic expansions at these points. I can also construct Taylor ...

**20**

votes

**2**answers

2k views

### Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and
$$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$
See ...

**9**

votes

**2**answers

2k views

### Abel summation of the alternating series of primes?

Consider the ordinary generating function of the sequence of primes ($2+3x+5x^2+7x^3 + ...$); by the ratio test and the prime number theorem, its radius of convergence is $1$. Thus, we might well ask ...

**6**

votes

**1**answer

2k views

### Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection?

(This Question was taken from MSE. As Eric Naslund pointed out there, this question is relevant. The summation method mentioned in this question is actually a good answer to it.)
The Ramanujan ...

**4**

votes

**1**answer

330 views

### Is that series-transformation known in the context of divergent summation?

Note: I asked this question in math.stackexchange but did not receive an answer
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular ...

**4**

votes

**1**answer

494 views

### mertens-function in the light of divergent summation - what summation method were best adapted

Just reading about the Mertens-function in the other thread
Mertens function I remember an earlier attempt to apply divergent summation
to the series which is constructed of the Moebius-function
at ...

**7**

votes

**3**answers

851 views

### 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 ...

**33**

votes

**2**answers

2k views

### Is there a “quantum” Riemann zeta function?

Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the ...

**10**

votes

**5**answers

1k views

### Use of everywhere divergent generating functions

Generating functions are well-known to be much useful in combinatorics. But, maybe just since I am illiteral, all the applications coming in mind deal with power series, which are not just formal, but ...

**1**

vote

**3**answers

644 views

### 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 ...

**7**

votes

**3**answers

946 views

### Summing a divergent series and a constant combined

At least according to the answer to this question, $\zeta(1) = \gamma $ (once reqularized, of course).
Let me rephrase that by stating that:
$$ \sigma(\zeta(1)) = \gamma $$
Here, $\sigma(x)$ is the ...

**5**

votes

**3**answers

396 views

### Non-absolute convergence of series with asymtotically equal coefficients

The following seems to be a question related to standard calculus, but I am not quite sure
where to look for an answer.
Suppose $f,g:\mathbb{N} \to \mathbb{C}$ are such that the have the same ...

**1**

vote

**1**answer

436 views

### Testing for asymptotic series

I would like to know if there is a systematic way of writing down an asymptotic series representation of a function?
(like one can use Taylor series expansion for doing a power series).
Conversely ...

**15**

votes

**2**answers

805 views

### What's the cell structure of K(Z/nZ, 1)? Does it let me sum this divergent series? What about other finite groups?

The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be ...

**4**

votes

**1**answer

442 views

### Values of cusp forms at q = 1 ?

Take a cusp form $f$ and let $f(q) = q + a_2q^2 + q_3q^3 + \ldots$" denote its $q$-expansion (assume that the $a_k$ are integers, and that $f$ comes from an elliptic curve $E$). Of course the series ...

**23**

votes

**4**answers

2k views

### 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^{+}}
> ...

**20**

votes

**4**answers

4k views

### Summation methods for divergent series

There are many methods for assigning a value to a series that diverges, e.g. zeta function regularization, Abel summation, Cesaro summation, etc. From all of the examples I've found, two methods ...

**30**

votes

**8**answers

3k views

### Does any method of summing divergent series work on the harmonic series?

It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic ...