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**14**

votes

**0**answers

861 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 ...

**12**

votes

**0**answers

547 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**9**

votes

**0**answers

475 views

### Can an infinite sum depending on the logarithms of all positive integers be rational or algebraic?

Consider
$$ C = \sum_{n=1}^\infty \frac{(-1)^{f_1(n)} f_2(n) \log{n} + f_3(n)}{f_4(n)}$$
The sum converges. $f_1$ is either $0$ or $n-1$, $f_2,f_3,f_4$ are
polynomials with integer coefficients and $ ...

**8**

votes

**0**answers

189 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 ...

**8**

votes

**0**answers

402 views

### Composition of two formal series

There are two formal semi-infinite Laurent series
$$
f_+(z)=z+\sum_{k=2}^{\infty} a_k z^k
$$
and
$$
f_-(z)=z+\sum_{k=0}^{\infty} b_k z^{-k}
$$
Their composition (we assume that this composition ...

**7**

votes

**0**answers

289 views

### About the first decimal of $\sqrt {n!}$

Do we have :
$$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$
Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.

**6**

votes

**0**answers

173 views

### Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$
...

**6**

votes

**0**answers

128 views

### How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...

**6**

votes

**0**answers

271 views

### On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...

**6**

votes

**0**answers

212 views

### Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ...

**6**

votes

**0**answers

243 views

### Zeros of polynomials related to Jensen polynomial associated with Riemann xi function $\xi(x)$

We encountered polynomials defined by the recursive relations for the coefficients $b_k>0$ as defined below:
$$p_{n}(x)=\sum_{k=0}^{n}\binom{2n}{2k}b_k x^k$$
...

**6**

votes

**0**answers

357 views

### Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...

**5**

votes

**0**answers

121 views

### Are these identities Newton series?

Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...

**5**

votes

**0**answers

127 views

### Inverse problems for an asymptotic series which depends on a parameter?

I have the series
$\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$,
where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An ...

**5**

votes

**0**answers

365 views

### Number of Configurations in the optimal Hanoi tower

There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...

**4**

votes

**0**answers

335 views

### Elementary treatment of elementary functions in constructive math

I would appreciate a reference to constructive math literature with elementary proofs that elementary functions are locally non-constant (i. e. densely apart from any real in any interval with ...

**4**

votes

**0**answers

206 views

### Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas:
1) The ...

**4**

votes

**0**answers

169 views

### a question about Tsirelson's space

NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...

**4**

votes

**0**answers

219 views

### $\sum_{p,q \text{ primes } p \le q} 1/(pq\log(pq))$

The sum $$ \sum\limits_{p,q \text{ primes } p \le q} \frac{1}{pq\log(pq)}$$
is related to a conjecture of Erdős about primitive sequences.
It converges because the sequence is primitive. If my ...

**4**

votes

**0**answers

257 views

### Useful lower bound on an infinite sum

Fix integer $s.$ I have encountered the following infinite sum.
$$\sum_{k=0}^\infty \Pi_{l=1}^k\left[1-(1-2^{-l})^s\right]$$
Is there a useful lower bound on this expression? For instance, if $s=1,$ ...

**4**

votes

**0**answers

157 views

### Are there infinite sequences of rational cubes whose first differences are positive squares?

This is related to How many sequences of rational squares are there, all of whose differences are also rational squares?
Are there infinite sequences $a_n$ of rational cubes whose first ...

**4**

votes

**0**answers

202 views

### Number of times lead changes in a multi-candidate election (reference-request)

In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...

**4**

votes

**0**answers

689 views

### Computability of OEIS A034891 …partitions of n into prime parts (1 included)

On the seqfan mailing list RGWv gave short algorithm for computing A000041 number of partitions of n the partition numbers:
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == ...

**3**

votes

**0**answers

68 views

### Maximizing the discrepancy in Jensen's inequality for a certain function

Let $\underline{b}=\{b_1,\dots,b_n\}$ be a fixed sequence of positive numbers, and let $a>0$ be a parameter.
Define
$$
D(a;\underline{b}):=\frac{1}{\frac{1}{na}+\frac{1}{\sum_{i=1}^n b_i}}
...

**3**

votes

**0**answers

57 views

### Pointwise (a.e) evaluation of $\sum_{n \geq 0}(u,w_n)_{L^2}w_n$ and equalities in $L^2$

Let $w_n$ be a orthonormal basis of $L^2(\Omega)$. Given $u \in L^2$ we can write $$u=\sum_{n \geq 0}(u,w_n)_{L^2}w_n.$$
Suppose $w_n$ are the eigenfunctions of the Neumann Laplacian. We can write
...

**3**

votes

**0**answers

107 views

### Euler series with milder divergence

Theorema 19 in Euler's memoir "Variae observationes circa series inﬁnitas" says
The sum of the reciprocals of the prime numbers is inﬁnitely great but is inﬁnitely times less than the sum of the ...

**3**

votes

**0**answers

233 views

### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...

**3**

votes

**0**answers

113 views

### The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...

**3**

votes

**0**answers

171 views

### Do $r$-th root Harmonic numbers ever sum to integers?

None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$
(e.g., this MSE question and answer).
Q. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n ...

**3**

votes

**0**answers

99 views

### On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset ...

**3**

votes

**0**answers

117 views

### “Shifted” Vandermonde determinant is nonzero?

I have already posted this question at MSE here, but as it received a few upvotes, but no comments or answers I choose to cross-post it here.
Let $P$ be a degree-two polynomial, with roots ...

**3**

votes

**0**answers

112 views

### Prove that when converge, the following expansions are equal

Prove $f_1(x)=f_2(x)=f_3(x)$ when converge.
$$f_1(x)=\sum_{m=0}^{\infty} \binom {x}m \sum_{k=0}^m\binom mk(-1)^{m-k}f(k)$$
$$f_2(x)=\lim_{n\to\infty}\binom xn\sum_{k=0}^n\frac{x-n}{x-k}\binom ...

**3**

votes

**0**answers

125 views

### A second polylogarithm ladder for the tribonacci and n-nacci constants

In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with,
$$0 = ...

**3**

votes

**0**answers

307 views

### The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is ...

**3**

votes

**0**answers

147 views

### Shift-invariant submultiplicative seminorms of $\ell^{\infty}$

Question: Is there a shift-invariant submultiplicative seminorm $||\cdot||$ of $\ell^\infty$ which satisfies the following property?
If $f:\mathbb{N}\rightarrow\mathbb{N}$ is an increasing function ...

**3**

votes

**0**answers

267 views

### Convolution inverse of recursively defined sequence is alternating

Consider the double sequence $A(n,k)$ which is recursively defined by
$$A(n,n)=1 \text{ for } n=0,1,2,\dots \text{ and }$$
$$A(n,k)=2\sum_{l=1}^{k+1} \binom{2n+1}{2l} A(n-l,k+1-l) \text{ for }0\leq k ...

**3**

votes

**0**answers

229 views

### Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely
$$ \small f_p(x) = \sum_{k=0}^{\infty} ...

**3**

votes

**0**answers

727 views

### Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation?
For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...

**3**

votes

**0**answers

328 views

### Finch's sequence over $\mathbb{F}_3$

In http://algo.inria.fr/csolve/seqmod3.pdf -- "Periodicity in sequences mod 3" Steven Finch (also cited in Sloane's OEIS A112683) defines the following sequences in $\mathbb{F}_3$:
For each positive ...

**2**

votes

**0**answers

87 views

### Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...

**2**

votes

**0**answers

201 views

### Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences.
Let $p \in ...

**2**

votes

**0**answers

260 views

### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?

**2**

votes

**0**answers

140 views

### Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture
It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...

**2**

votes

**0**answers

205 views

### Proving that an increasing iterative sequence increases at a decreasing rate

In this question
Proving a sequence of integrals increases (iterated minimax distributions)
Pietro Majer proved that
$$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = ...

**2**

votes

**0**answers

442 views

### How to calculate/approximate expectation of function of a binomial random variable?

Hi,
I am stuck at following problem in my research.
Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. ...

**2**

votes

**0**answers

604 views

### What square-summable sequences are “sinc-summable”?

$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} ...

**2**

votes

**0**answers

507 views

### recursive sum of products of bessel functions

I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$
recursively, i.e. once $S_0(x,y)$ is found, via the ...

**1**

vote

**0**answers

97 views

### Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then
$$
f(x)=\sum_{n\geq 0}a_n x^n
$$
converges absolutely for all $x$. Under ...

**1**

vote

**0**answers

152 views

### Can one show that the terms of prime index of a certain recursive sequence are not divisible by their index infinitely often?

Let $(a_n)$ be the sequence defined by
$$a_{n+1}=2na_n-n^2a_{n-1}$$
and $a_0=0$ and $a_1=1$. I would like to prove that there exist infinitely many primes $p$ such that $p$ does not divide $a_p$. Any ...

**1**

vote

**0**answers

105 views

### Four kinds of generalized hypergeometric formulas for $\pi$

Given,
$$\begin{array}{|c|c|c|c|}
\hline
n&a_n&b_n&c_n\\
\hline
1 & 6541681608 & 163096908 & -640320^3\\
\hline
2 & 85840 & 4492 & -14112^2\\
\hline
3 & 28302 ...