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

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### A nonlinear initial-boundary value problems with Taylor expansion of parameter [closed]

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem
$$
u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1
$$
$$
u(0,t) = 0 \\
u(1,t) = 0 \\
u(x,0) = \epsilon f(x) ...

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### Does this sequence always give an integer?

It is known that the $k$-Somos sequences always give integers for $2\le k\le 7$.
For example, the $6$-Somos sequence is defined as the following :
$$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot ...

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**2**answers

507 views

### $ - \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1}$ [closed]

Numerical evidence suggests:
$$ - \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1} \approx 0.764499780348444 $$
Couldn't find cancellation via rearrangement.
For the ...

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**2**answers

453 views

### Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence?

Question : Letting $k,n$ be positive integers, let's define a sequence $\{a_i\}\ (i=0,1,\cdots, kn)$ as
$$(1+x+\cdots+x^k)^n=\sum_{i=0}^{kn}a_ix^i.$$
Then, is the 'special' difference-sequence ...

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215 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 ...

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**0**answers

81 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}}.
...

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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 = ...

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355 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 ...

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**0**answers

474 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 $ ...

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**1**answer

510 views

### Is this known alternating sum for Euler's constant?

This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...

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**1**answer

271 views

### Lambert series identity

Can someone give me a short proof of the identity,
$$\sum_{n=1}^\infty\frac{q^nx^{n^2}}{1-qx^{n}}+\sum_{n=1}^\infty\frac{q^nx^{n(n+1)}}{1-x^n}=\sum_{n=1}^\infty\frac{q^nx^n}{1-x^n}$$

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**1**answer

206 views

### non-trivial convergent sequence [duplicate]

I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...

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**1**answer

208 views

### A question on graphic sequences

Let $G$ be a graph and $d_{G}(u)$ denotes degree of a vertex $u$ in $G$. Consider the next multiset $$M_{G}:=\{|d_{G}(u)-d_{G}(v)|:\ uv\in E(G)\}.$$
Conjecture: $M_{G}$ is graphical for every $G$.
...

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**2**answers

1k views

### Numbers that are generic w.r.t. exponentiation

This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways.
...

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**1**answer

228 views

### A complex sequence with positive values

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d ...

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**1**answer

1k views

### Beyond Collatz: A $5n+1$ conjecture? [closed]

Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...

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### Convergent subsequence of $\sin n$

It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such ...

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262 views

### Certain asymptotics involving double infinite sum

Let $1<\alpha<\beta<3/2$. Set
$$
S(n)= \sum_{i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}.
$$
One can check that $S(n)$ is finite. My question is when $n\rightarrow ...

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**1**answer

253 views

### upper bound on infinite series by exponentials

I'm interested in upper bounding the following infinite series by exponential functions for x>1 (I would be fine with x>2 as well).
\begin{eqnarray*}
\sum_{n=1}^{\infty}e^{-\frac{x^2n^2}{2}}\\
...

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**3**answers

433 views

### sequences of real numbers

Let $\lbrace x_i\rbrace_{i=1}^\infty$ be a sequence of distinct numbers in $(0,1)$. For any $n$ after deleting $x_1,...,x_n$ from $[0,1]$ we get $n+1$ subintervals. Let $a_n$ be the maximum length of ...

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**1**answer

526 views

### Busy Beaver modulo 2

There is well-known Rado's "Busy Beaver" sequence — the maximal number of marks which a halting Turing machine with n states, 2 symbols (blank, mark) can produce onto an initially blank two-way ...

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**1**answer

272 views

### Questions about expansion of $f(x)=\sum_{i=1}^{\infty} a_i x^i$

In complex field, assume $f(x)=\sum_{i=1}^{\infty} a_i x^i$ where $a_i \in {\bf N}$ or $a_i = 0$, and $f(x)$ converges in an area.
Question 1: are there $$f(x)=p(x)+\sum_{i=1}^{\infty}r_i(x), $$
or ...

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### Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?

I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.
...

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741 views

### An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...

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105 views

### Terminology for sequences/functions that approach each other

What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are ...

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323 views

### How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$?

I have to estimate the expression $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for $\kappa$ very small $\kappa \sim 10^{-19}$ and $N$ very large $N\sim 10^{26}$ and $a$ arbitrary $a=1, \ldots, N$. I do not ...

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375 views

### A Sequence of Real numbers

Consider the sequence $\lbrace \frac{\phi(i)}{i}\rbrace_{i=1}^\infty$ where $\phi$ is the Euler's function. The Sequence is clearly dense in $[0,1]$. What can be said about the limsup of its average ...

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301 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 ...

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**1**answer

91 views

### Series of terms following polynomial recurrence relation

Hello!
I'm stuck trying to prove a statement that seems very reasonable based on computer experiments.
For an integer $N$ and any $\epsilon \in (0,1)$, I define a sequence $u_0 = N$ and for all ...

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543 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 ...

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**2**answers

383 views

### How this expression leads to the given sequence

Here given is a sequence from OEIS.
The sequence is triangle of coefficients from fractional iteration of $e^x - 1$. Few terms are:
1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, ...

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**0**answers

230 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 ...

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**4**answers

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### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

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**2**answers

1k views

### On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$.
I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...

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**0**answers

275 views

### Infinite Sum $\sum_{n=1}^\infty 2x\arctan(x/n)-n\log(1+x^2/n^2)$

I ran into the infinite sum $\sum_{n=1}^\infty 2x\arctan(x/n)-n\log(1+x^2/n^2)$, where $x$ is a positive real number. Mathematica can't do the sum, but shows that it's very well approximated by ...

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737 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

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**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 ...

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**1**answer

233 views

### An elementary expression for $_3F_2(1,1,9/4;2,2;-1)$

Consider the following series:
$$S=\sum_{n=1}^\infty\frac{(-1)^n\ \Gamma\left(\frac{5}{4}+n\right)}{n^2\ \Gamma(n)}.$$
It can be expressed in terms of a hypergeometric function:
...

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**5**answers

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### Proving that every term of the sequence is an integer

Let $m,n$ be nonnegative integers.
The sequence $\{a_{m,n}\}$ satisfies the following three conditions.
For any $m$, $a_{m,0}=a_{m,1}=1$
For any $n$, $a_{0,n}=1$
For any $m\ge0, n\ge1$, ...

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**1**answer

308 views

### Upper bound of a series

Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality:
$$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$

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127 views

### Series expansion with remaining $log n$

Hi,
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a ...

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**2**answers

304 views

### A sequence based on Catalan–Mihăilescu problem

It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The conjecture was ...

**0**

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**1**answer

92 views

### positive expression

Let
$$a_{n,k}=\sum_{s_i \geq 1 \atop \sum_{i=1}^{n-k} s_i \leq n} \frac{2^{n}}{(2(n-\sum_{i=1}^{n-k} s_i)+1)!\prod_{i=1}^{n-k} (2s_i)! }$$
for $0 \leq k \leq n-1$. Prove for $1 \leq k \leq n-1$ that
...

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**1**answer

402 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 ...

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**1**answer

584 views

### To express $e^{\sum \limits_{k=0}^\infty q^{2^k}}$ as product terms of $(1-q^k)^{c(k)}$

$|q|\lt1$
$A(q)=\sum \limits_{k=0}^\infty q^{2^k}$
Easily We can see that
$$A(q)=q+A(q^2)\tag 1$$
Let's assume we redefine $A(q)$ as below
$A(q)=-\sum \limits_{k=1}^\infty c_k \ln{(1-q^k)}$
I ...

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**0**answers

364 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 ...

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122 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 ...

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265 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 ...

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69 views

### Equally subspacing the support of a monotone function, maintaining its mean

SETUP:
Assume $f(\cdot)$ is continuous and strictly monotone decreasing on $[0,\infty]$, with $f(0)>0$ and $f(\infty)<0$.
Let $x_m$ be the solution of $\frac{1}{m}\sum_{i=1}^{m}f(ix)=0$, where ...

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**1**answer

291 views

### Optimality of p-Lebesgue Differentiation Theorem for Sobolev Functions

This is the third question in a series whose purpose has been to flesh out an example of the optimality of the p-Lebesgue differentiation theorem for Sobolev functions. This theorem says that for $f ...