All Questions
Tagged with divergent-series sequences-and-series
54 questions
2
votes
1
answer
254
views
New (?) Regularization Method for Divergent Series [closed]
Playing with identities ($1$) and ($2$) from this blog post and infinite geometric series, I've noticed the following.
For $x > 1$, the following series is convergent:
$$\sum_{n=0}^{\infty} e^{(2n ...
2
votes
0
answers
209
views
A problem about the series $\sin(n^p)$ [closed]
Prove that when $p>0,$ the series $$\sum_{n=1}^\infty \sin(n^p)$$
is divergent
- 67
4
votes
1
answer
418
views
Series convergence if $\sum a_n^2 < \infty$
There are quite a few simple results about convergent/divergent series derived from similar ones. Here is a question in the same spirit that I saw posted on another forum. Unfortunately, I don't have ...
- 689
25
votes
1
answer
2k
views
Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
- 2,689
2
votes
2
answers
260
views
Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,730
7
votes
1
answer
337
views
If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
- 1,730
6
votes
0
answers
171
views
Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
- 1,730
4
votes
0
answers
160
views
Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
-2
votes
1
answer
217
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 317
6
votes
1
answer
241
views
Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
- 1,730
9
votes
0
answers
313
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. ...
- 1,730
7
votes
0
answers
306
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 ...
- 1,730
3
votes
0
answers
171
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^...
- 1,730
21
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
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 1,730
1
vote
1
answer
117
views
On summation methods of divergent series
$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\CC}{\mathcal C}$This previous question introduced the following notion of a summability space.
Let $\N:=\{1,2,\...
- 128k
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
5
votes
3
answers
343
views
Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 2,689
1
vote
1
answer
2k
views
Interchange summation order in the limit of number of elements going to $\infty$
Considering the sum $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty} a_{ij}$, in general we are not allowed to interchange the summation order (i.e. pass to $\sum_{j=0}^{\infty}\sum_{i=0}^{\infty} a_{ij}$) but ...
- 305
0
votes
1
answer
271
views
Solving (or approximating) a certain delay differential equation
I'm interested in finding the (unique?) solution to the set of delay differential equations
$$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^...
- 1,730
3
votes
1
answer
281
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 ...
- 19.7k
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$
3
votes
0
answers
276
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 ...
- 1,730
3
votes
2
answers
461
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 ...
- 1,730
12
votes
1
answer
1k
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 ...
- 1,730
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, ...
- 1,730
2
votes
1
answer
247
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 ...
- 1,730
4
votes
1
answer
205
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$$
converges and the limit $\lim_{x \to 1^{-}} g(x)$ exists....
- 2,404
3
votes
0
answers
79
views
Some exercise on the regularity of a summability method
I was reading the book of Johann Boos "Classical and modern method in summability theory" and I came across an exercise from the Chapter 2 (page 50, exercise 2.3.15). Here is the statement ...
- 31
3
votes
1
answer
285
views
Is there an asymptotic bound between converging and diverging series? [closed]
Let us define for every $k\in\mathbb{N}$ and every large enough $x\in \mathbb{R}$,
$$
\log^{[k]}(x) =
\begin{cases}
\log^{[k-1]}(\log(x)) & k>0 \\
x & k=0
\end{cases}.
$$
It is well known, ...
- 41
6
votes
0
answers
2k
views
Do smooth cutoff functions analytically continue functions?
My goal is to prove (or disprove) that sufficiently smooth and quickly decaying cutoff functions being tacked on to a Taylor series correctly extend the radius of convergence to the analytic ...
- 1,730
2
votes
0
answers
232
views
Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences.
Still, in my view there is fundamental difference between divergent ...
- 10.1k
4
votes
0
answers
289
views
Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$
Here $\mu(n)$ is Möbius function and $M(x)$ is Mertens function.
The computations show that the partial sums $\sum_{n \le x} \frac{\mu(n)}{\sqrt{n}}$ stays between $-0.2$ and $-1.2$ when $10^1<x<...
- 203
4
votes
1
answer
2k
views
Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
- 91
2
votes
2
answers
385
views
What is the growth rate of the sum of powers of distinct primes closest to a given a integer?
Let $n$ be a positive integer, and
$$2 = p_1 < p_2 < \dots < p_m \le n$$
be the sequence of all primes less than or equal to $n$.
For each index $j$ let $p_j^{e_j}$ be the largest power of $...
- 557
1
vote
1
answer
88
views
Convergence properties of related series
Let $u_m = \ln ^2 m$.
Does there exist a non-increasing sequence of positive numbers $\{g_n\}_{n \in \mathbb{N}}$, $g_n \to 0$, such that
$$\sum\limits_{n \in \mathbb{N} } g_n = \infty, \ \ \ \ \...
- 724
7
votes
2
answers
976
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 ...
- 2,203
1
vote
2
answers
604
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:
...
- 2,203
3
votes
1
answer
184
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 ...
1
vote
1
answer
270
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\...
- 9
0
votes
0
answers
89
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 ...
- 109
2
votes
0
answers
109
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 ...
- 10.1k
2
votes
0
answers
143
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 ...
- 1,150
0
votes
2
answers
188
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 ...
- 416
1
vote
2
answers
115
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) \...
- 195
12
votes
1
answer
742
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 ...
- 4,829
13
votes
1
answer
782
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 ...
- 5,342
6
votes
1
answer
644
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 n/\Phi}(p ...
- 241
3
votes
1
answer
578
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 ...
- 5,342
6
votes
1
answer
454
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 $$
where I want to make ...
- 5,342
16
votes
0
answers
1k
views
Regularizing the divergent sum $1^k + 2^k + \cdots$
EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...
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