Questions tagged [divergent-series]

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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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2 votes
0 answers
65 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^...
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4 votes
1 answer
102 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 ...
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5 votes
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261 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 ...
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2 votes
1 answer
195 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 ...
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2 votes
0 answers
88 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 $L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2$
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2 votes
0 answers
96 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 ...
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3 votes
0 answers
129 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 ...
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3 votes
2 answers
285 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 ...
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10 votes
1 answer
742 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 ...
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8 votes
2 answers
1k 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, ...
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1 vote
1 answer
156 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 ...
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4 votes
1 answer
156 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....
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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 ...
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3 votes
1 answer
205 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, ...
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6 votes
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602 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 ...
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1 vote
0 answers
80 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 ...
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2 votes
0 answers
192 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 ...
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4 votes
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232 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<...
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3 votes
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394 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. $$\...
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2 votes
0 answers
259 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 ...
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10 votes
2 answers
863 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 $\sum_{n\in A} n^{-1}$ ...
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4 votes
1 answer
831 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 ...
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2 votes
2 answers
232 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 $...
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1 vote
1 answer
71 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, \ \ \ \ \...
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1 vote
0 answers
57 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 ...
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3 votes
2 answers
276 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 ...
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6 votes
3 answers
436 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 ...
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7 votes
2 answers
690 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 ...
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1 vote
1 answer
409 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 ...
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1 vote
2 answers
467 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: ...
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1 vote
1 answer
93 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 ...
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4 votes
0 answers
911 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-...
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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 ...
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3 votes
1 answer
164 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 ...
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0 votes
0 answers
102 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) ...
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1 vote
1 answer
224 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\...
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3 votes
0 answers
348 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^\...
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1 vote
1 answer
184 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 ...
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2 votes
0 answers
186 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/...
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-1 votes
2 answers
676 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 ...
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0 votes
0 answers
80 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 ...
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2 votes
0 answers
102 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 ...
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1 vote
0 answers
203 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 ...
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-1 votes
1 answer
168 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 ...
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0 votes
1 answer
171 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 ...
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0 votes
1 answer
82 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 ...
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6 votes
1 answer
266 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}...
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  • 171
2 votes
0 answers
138 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 ...
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3 votes
2 answers
305 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 ...
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