The sequences-and-series tag has no wiki summary.

**14**

votes

**0**answers

796 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

515 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 ...

**8**

votes

**0**answers

466 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

375 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

137 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 ...

**6**

votes

**0**answers

167 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

194 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

319 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

220 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 ...

**5**

votes

**0**answers

121 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

334 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 ...

**5**

votes

**0**answers

316 views

### Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...

**4**

votes

**0**answers

147 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

153 views

### Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} +
...

**4**

votes

**0**answers

212 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

219 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

150 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

195 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

678 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

151 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

91 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

96 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

106 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

215 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

138 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

258 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

217 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

674 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

319 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

164 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

102 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 = ...

**2**

votes

**0**answers

125 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

188 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

370 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

484 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

461 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

43 views

### Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here.
In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...

**1**

vote

**0**answers

50 views

### Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here.
Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...

**1**

vote

**0**answers

112 views

### Reference for a General Theory of Sequences?

Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis.
In that works, sequence spaces are generally ...

**1**

vote

**0**answers

101 views

### Passing to the limit in a PDE (subsequence problems)

For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...

**1**

vote

**0**answers

279 views

### Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ was derived here:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(\sum _{n=1}^{\infty } {\frac {s-1-2\,n}{{n}^{s}}} + ...

**1**

vote

**0**answers

82 views

### A generalization of alternating series involving modulus?

Alternating series are common in the literature, with important examples including
$\displaystyle\sum_{n=1}\frac{(-1)^{n-1}}{n}=\log 2$,
...

**1**

vote

**0**answers

86 views

### Structural differences between closed forms of two related infinite products?

In this question, I went quite a bit over the top, so I now tried to rephrase it in a much simpler way.
Take $a \in \mathbb{R}, s \in \mathbb{C}$ and:
$$\displaystyle C(s,a) := \prod_{n=1}^\infty ...

**1**

vote

**0**answers

195 views

### Convergence of $\sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$

Related to an open problem about another series.
Set
$$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$
where $\psi^{(n)}(k)$ is the polygamma function.
Does $A$ converge?
The related ...

**1**

vote

**0**answers

76 views

### The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion:
\begin{equation*}
B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}.
...

**1**

vote

**0**answers

189 views

### Zeta sum $\sum_{n=2}^\infty \frac{\zeta(n)}{a^n}$

Probably this is known, but mathworld and wolfram alpha don't
recognize this potential identities.
Numerical evidence suggests:
$$ \sum_{n=2}^\infty \frac{\zeta(n)}{a^n} =? \sum_{n=1}^\infty ...

**1**

vote

**0**answers

107 views

### Bounding a recursively defined sequence

I have a sequence $\lambda_0,\lambda_1,\ldots,$ which is defined recursively as
$$ \lambda_0 = \frac{1}{2},$$
and
$$\lambda_{k+1} = \max_{\lambda\in [1,b]} \left(\frac{1}{2\lambda}\prod_{0\leq ...

**1**

vote

**0**answers

162 views

### Calculating $n$ for $\sigma(\sigma(n)-n) = n$ [redefined]

As in A072868 described by OEIS;
Defined by $\sigma(\sigma(n)-n) = n$.
Since these numbers are important in regard to many things, specially mersenne primes, since ${n-1 \over 2}\times ...

**1**

vote

**0**answers

134 views

### Limit of Sequence of unusual Prime Product

Let $p_n$ be the nth prime and $p_L$ be closest to its square root:
\begin{equation}
p_L^2 \approx p_n \approx x
\end{equation}
Let $\sigma \in Z^+$ be a positive integer constant. Define the ...

**1**

vote

**0**answers

207 views

### Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?

This is somewhat related to Erdős conjecture on arithmetic progressions
Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ ...