Questions tagged [divergent-series]
The divergent-series tag has no usage guidance.
131 questions
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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-...
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150
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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 ...
- 63.9k
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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 $1^n+2^n+3^n+4^n+\cdots$
A value can also be assigned to the related series ${n+0 \choose n}+{n+...
CommunityBot
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1
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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 ...
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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, ...
- 2,836
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6
answers
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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 ...
- 18.4k
2
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1
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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 ...
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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....
- 91.8k
3
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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 ...
- 1,848
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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, ...
- 6,367
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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
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0
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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 ...
- 10.1k
2
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0
answers
232
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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 ...
- 10.1k
3
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1
answer
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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 ...
- 41.1k
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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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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.
$$\...
- 10.1k
2
votes
0
answers
280
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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 ...
- 10.1k
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5
answers
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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 ...
- 455
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2
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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}$ ...
- 105k
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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 ...
- 2,902
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1
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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 ...
- 63.9k
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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 ...
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2
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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 $$
Neither of these questions obtained a ...
- 18.4k
2
votes
2
answers
385
views
What is the growth rate of the sum of powers of distinct primes closest to a given a integer?
Let $n$ be a positive integer, and
$$2 = p_1 < p_2 < \dots < p_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p_j^{e_j}$ be the largest power of $...
- 6,984
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vote
1
answer
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views
Convergence properties of related series
Let $u_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g_n\}_{n \in \mathbb{N}}$, $g_n \to 0$, such that
$$\sum\limits_{n \in \mathbb{N} } g_n = \infty, \ \ \ \ \...
- 24.2k
1
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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 ...
- 11
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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 ...
- 2,203
1
vote
1
answer
415
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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 ...
- 10.1k
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2
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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 ...
- 10.1k
1
vote
1
answer
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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 ...
- 24.8k
2
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1
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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 ...
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3
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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 ...
- 188k
1
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1
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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 $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
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Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)
(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...
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New/useful method for summation of divergent series?
Questions
$$ 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^\...
- 237
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1
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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 ...
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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 ...
- 10.1k
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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/...
- 10.1k
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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 ...
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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 ...
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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 ...
- 10.1k
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1
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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 ...
- 5,342
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0
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280
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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 ...
- 11
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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 ...
0
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1
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220
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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 ...
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2
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Dimensional regularization in odd dimensions
I am not quite sure that my question below is appropriate for this site, probably it should be addressed to the physical commutity. But I hope that some (mathematical) physicists do attend MO. I have ...
- 1,996
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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 ...
- 1,228
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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 zeta-...
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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 ...
- 91.8k
6
votes
1
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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}...
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