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18
votes
1answer
382 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 ...
1
vote
1answer
268 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 ...
10
votes
3answers
551 views

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. ...
12
votes
2answers
725 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 ...
5
votes
1answer
101 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 ...
4
votes
2answers
314 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 ...
5
votes
1answer
366 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 ...
3
votes
0answers
208 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 ...
4
votes
1answer
87 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 ...
12
votes
0answers
514 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 ...
1
vote
2answers
326 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, ...
1
vote
0answers
186 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 ...
20
votes
4answers
1k views

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 ...
5
votes
3answers
729 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 ...
0
votes
0answers
223 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 ...
23
votes
1answer
681 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 ...
3
votes
0answers
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 ...
11
votes
1answer
191 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: ...
15
votes
5answers
1k views

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 $m\ge0, n\ge1$, ...
-1
votes
1answer
188 views

Upper bound of a series

Given $N$ and $a$ positive integers, with $a\ge 2$ is it possible to prove the inequality: $$\sum_{k=1}^N\frac{k^a}{(k+1)^a+(k+2)^a}\le\frac{N}{2}$$
0
votes
0answers
115 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 ...
12
votes
2answers
286 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
votes
1answer
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 ...
3
votes
1answer
370 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 ...
2
votes
1answer
460 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 ...
5
votes
0answers
333 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 ...
1
vote
0answers
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 ...
3
votes
0answers
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 ...
0
votes
0answers
68 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 ...
4
votes
1answer
269 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 ...
0
votes
2answers
182 views

sequence, such that sum of any combinations in the sequence does not equal another [closed]

Hi, Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence. ...
0
votes
1answer
195 views

Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$. Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise Let ...
6
votes
1answer
356 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 ...
1
vote
1answer
145 views

Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then ...
1
vote
1answer
406 views

Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would ...
2
votes
1answer
132 views

A series question related to solution of Laplace equation

Let $u(x,y)$ be the solution of the Laplace equation $\Delta u=0$ on the unit square $(0,1)\times (0,1)$ with boundary condition: $$ u(x,1)=1, u(x,0)=0, u(0,y)=0, u(1,y)=0$$ The series solution is ...
1
vote
3answers
2k views

$\sum _{k=0}^{\infty } \frac{1}{(k+m) k!} \equiv 1$ for $m=2$

I changed the title and added revisions and left the original untouched For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the ...
1
vote
1answer
136 views

Distorted Newtion binomial

This is a cross-posting of a MSE question (which did not receive any feedback there so far). Let $\varepsilon >0$, with $\varepsilon \neq 1$. Consider the sequence $u_n$ defined by $$ ...
3
votes
1answer
256 views

Does there exist a sequence of complex numbers such that…

The following question came up while I was working through an example: Does there exist an $\ell^1$ sequence of complex numbers $a_n$, not all zero, such that $\sum_n a_n n^{-p} = 0$ for all $p ...
13
votes
1answer
2k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
1
vote
0answers
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
1answer
414 views

Limit of functions and asymptotic behaviour

Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$. I denote the polynomials $p_l(t) = \sum_{k=0}^{m(l)} ...
22
votes
1answer
383 views

“Harmonacci” recurrence and identities for $\pi$

While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$ It turns out that given $a_0 = 1, a_1 = 1$, $$lim \frac{a_{2n}}{a_{2n-1}} = ...
1
vote
1answer
76 views

Series of quotients with perturbed denominator

Let $a_n>0$ and $b_n>0$ be two strictly declining sequences such that the series $$\sum_{n=1}^\infty \frac{a_n}{b_n}$$ is convergent. For $\sigma>0$ define $$f^N(\sigma) = \sum_{n=1}^N ...
3
votes
1answer
380 views

What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...
3
votes
2answers
278 views

Given a sequence of real numbers,do the following conditions suffice to guarantee convergence to 0?

If $x_{a+1}$-$x_{a}$ converges to $0$ and $x_{2a}$-$2x_{a}$ converges to $0$ , does that imply $x_a$ converges to $0$?
0
votes
1answer
99 views

Solution of certain forms of equations

I ask about a possible method to find the solution of algebraic equations of the form $axⁿ+byⁿ+c=0$ where $a,b,c,x,y$ are real constants and $n$ is an integer. Maybe there is a simple method, but I ...
1
vote
0answers
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 ...
6
votes
1answer
962 views

Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
4
votes
2answers
691 views

series expansion of the q-Pochhammer symbol

The following identity arose while I was working on a recent MO question: $-\sum_{n=1}^{\infty}\frac{1}{n}\frac{(-x)^n}{1-x^n}=\sum_{n=1}^{\infty}\frac{1}{n}\frac{x^n}{1-x^{2n}}.$ I have no doubt ...