Tagged Questions
2
votes
3answers
71 views
What type of Hypergeometric series is this?
I am trying to find a closed form for the series
$$ \sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right)$$
$m$ is a nonzero positive integer, and $b$, ...
41
votes
3answers
648 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
...
6
votes
1answer
95 views
Umbral calculus with negative indices (and powers)
Can we do umbral calculus with negative indices (and powers)? Can we write $a_{-n} \equiv a^{-n}$ or $L[a_{-n}] = a^{-n}$ where $L$ is a linear functional and $n$ need not be negative?
The common ...
1
vote
1answer
28 views
Searching for a certain sequence
How do I search OEIS for a certain sequence? For example I want the number of ways $1, 2, 3,\ldots, n$ can be arranged so that the numbers in the even places are greater than their neighbours.
3
votes
1answer
74 views
series involving $\log \left(\tanh\frac{\pi k}{2} \right)$
I found an interesting series
$$\sum_{k=1}^\infty \log \left(\tanh \frac{\pi k}{2} \right)=\log(\vartheta_4(e^{-\pi}))=\log \left(\frac{\pi^{\frac{1}{4}}}{2^{\frac{1}{4}}\Gamma \left( ...
12
votes
3answers
285 views
Has someone seen a discussion of the (divergent) summation of $\sum\limits_{k=0}^\infty (-1)^k (k!)^2 $?
In extending my studies of the Eulerian matrix and its suitability for a matrix-based divergent summation procedure I'm trying to proceed to sums of the form
$$ S = \sum_{k=0}^\infty (-1)^k (k!)^2 $$
...
3
votes
2answers
86 views
Sequence and series problems with slick group theoretic solutions.
So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and ...
8
votes
2answers
171 views
How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
I'm considering the transfer-function
$$ t(x) = \log(1 + \exp(x)) $$
and find the beginning of the power series (simply using Pari/GP) as
$$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
0
votes
0answers
10 views
Qualifying Parameters
you have two parameters, 1) rates of trees per land size, ranging from 30%-100%, and 2) rates of birds per land size, ranging from 5%-30%
goal is that you're trying to find out which is overall ...
4
votes
1answer
191 views
A theorm about Cesàro mean, related to Stolz-Cesàro theorem
Original Title: Tauberian theorems and Cesàro sum
Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)
$\newcommand\abs[1]{\left\lvert#1\right\rvert}$
If $\{s_n\}$ ...
0
votes
0answers
30 views
Resources for learning Series.
First, I am studying for the AP BC Caclulus Exam, and I don't have much on series to study from; however, I would like some resources that extend the knowlege beyond the basics. I notice that series ...
2
votes
0answers
67 views
Double sequences and series double.
I would like a good reference devoted exclusively to double sequences and series double. At the moment I have only this reference in google.
Thanks in advance.
4
votes
1answer
78 views
A reference for some fact in analysis
I am looking for a reference for the following fact. Any hints would be appreciated.
Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
0
votes
1answer
55 views
Convergence of series of elementary symmetric functions
Let $x_1,x_2,x_3,\ldots$ be an infinite sequence of real numbers (or assume they're complex numbers if you find that convenient).
Let $e_0,e_1,e_2,e_3,\ldots$ be the elementary symmetric functions of ...
2
votes
0answers
46 views
tangents of infinite sums — reference request
This section of Wikipedia's List of trigonometric identities was, as far as I know, written entirely by me. For a time, it dealt only with finite sums, and said something like this:
$$
...
1
vote
2answers
67 views
About the sequence $\{\sin (n^2) \}_n$
A colleague of mine asked me if the behavior of the sequence $\{\sin (n^2) \}_n$ is known. In particular, does it converge? If not, what are its liminf and limsup?
I had to admit that I cannot answer ...
1
vote
1answer
34 views
How to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?
Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?
For example, I might want to approximate
$$
...
10
votes
0answers
172 views
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty ...
7
votes
2answers
113 views
How to find convergence region of $\sum_{n\geqslant 0, m \geqslant 0} x^n y^m \binom{n+m}{n}^2$
The following two series are special cases of Appell $F_3$ and $F_4$, namely:
$$
\mathcal{S}_1 = \sum_{n \geqslant 0, m \geqslant 0} \frac{x^n y^m}{\binom{n+m}{n}}
$$
and
$$
\mathcal{S}_2 = ...
1
vote
0answers
74 views
Double series info
Do you know some dedicated books for the double series manipulations and examples with them? I need some recommendation on the subject. I noticed
that the manipulations with double series are very ...
10
votes
1answer
221 views
What and where in the notebooks of Ramanujan is this series?
The wikipedia page on Ramanujan contains the following series:
$$
1 - 5\left(\frac{1}{2}\right)^3 + 9\left(\frac{1\times3}{2\times4}\right)^3 - ...
8
votes
2answers
121 views
Seeking analytic proof for $\sum_{n=r}^\infty \frac{1}{n!}\left[ n-1 \atop r-1 \right] = 1$
In Blom, Holst, Sandell, "Problems and snapshots from the world of probability", section 9.4, a model of records is discussed:
Elements are ordered in a sequence of increasing length according to ...
1
vote
2answers
94 views
Problem regarding infinite sum of remainders.
Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly):
Find
...
4
votes
4answers
219 views
Divergent series and $p$-adics
If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct.
Surely ...
24
votes
3answers
1k views
Asymptotic (divergent) series
MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in ...
1
vote
2answers
85 views
What is the list of theorem that are able to find out a sequence is converge or not?
A sequence is called converge if for every next term of the sequence is getting closer to the limit of a number.
What is the list of theorem that are able to helping to find out a sequence is converge ...
5
votes
0answers
76 views
Determinant expression for the power sum
Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression:
$$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
3
votes
2answers
187 views
What is this generalized multivariable hypergeometric function?
I'm looking for any kind of reference on a multivariable generalization of a (confluent) hypergeometric function.
To be specific, Horn's List is a list of 34 two-variable hypergeometric functions, 20 ...
3
votes
1answer
85 views
Sequences convergent to 'cycles'
Consider sequences $(x_n)_{n=1}^\infty\subset\mathbb R$. Is there a name for the following property?
There exists $L\in\mathbb N$ such that:
...
1
vote
2answers
158 views
Something like : “recursive” harmonic numbers? Where can I read more?
In my other thread I discussed a matrix-decomposition; for one matrix (U) I found now a description of its entries, which may best be denoted as "recursive harmonic numbers". However, googling with ...
9
votes
2answers
201 views
Erdős: Sum of rational function of positive integers is either rational or transcendental
I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its ...
10
votes
2answers
311 views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of ...
7
votes
0answers
156 views
Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
I'm wondering about a simple question that has multiple possible variants depending on a few parameters. The prototypical one would be:
Does there exist an infinite series such that any two ...
8
votes
1answer
214 views
An infinite series involving the Zeta Function
I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants:
$$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$
and
...
6
votes
2answers
316 views
Reference request: Introduction to mathematical theory of Regularization
I asked the question "Are there books on Regularization at an Introductory level?" at physics.SE.
I was informed that "there is (...) a mathematical theory of regularization (Cesàro, Borel, ...
8
votes
1answer
401 views
Summation formula name
What is the name of the following summation formula?
$$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f - \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f'w,$$ where $w$ is the “sawtooth” function, defined by ...
2
votes
2answers
96 views
Estimating number of crossings for Erastothenes' Sieve
In this paper (2.1) I need to understand the formula for the total number of operations: $$\sum_{i=1}^{\pi(\sqrt n)}\frac{n}{p_i} \approx n\ln \ln n + O(n)$$ On a sidenote, since we're only checking ...
2
votes
0answers
131 views
Has anyone studied this particular sequence?
I'm going to tag this as reference request, since I'm mainly interested in finding out whether this kind of sequence has been named in literature before, just in order to acknowledge it for something ...
1
vote
0answers
125 views
Orthogonal polynomial interpolation of a function
I want to write down an arbitrary function $f$ as an (infinite) sum of orthogonal polynomials, e.g. for $f(x) = e^{\sin(x)}$, $f(x) = \sum{a_n T_n}$, where $a_n$ are the coefficients and $T_n$ are the ...
3
votes
1answer
119 views
Dirichlet series generating function, ordinary generating function
I am partly repeating my self here.
The sequence $a_n$ generated by the Dirichlet series generating function:
$\sum \limits_{n=1}^{\infty} \frac{a_n}{n^s}$
corresponding to $\zeta(s)^m$
seems to ...
2
votes
3answers
716 views
Good book for self-learning sequence and series
I would be very happy if it covers sequence and series from very basics to advanced. Thanks.:)
3
votes
1answer
398 views
How to solve recurrence relations by the generalized hypergeometric series
I am reading methods of solving recurrence relation on Wikipedia. There is one method:
Many linear homogeneous recurrence
relations may be solved by means of
the generalized hypergeometric ...
1
vote
0answers
104 views
Is that series-transformation known in the context of divergent summation
Background:
In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ...
2
votes
1answer
173 views
General form for $(\sum\limits_{i=1}^n a_i)^n$
Does anybody know where I can find a general form, in terms of n, of the sum $\left(\sum\limits_{i=1}^n a_i\right)^n$. What I mean is, there appears to be some sort of pattern, if you look at ...
