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

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### 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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215 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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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 ...

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858 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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249 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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717 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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143 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

210 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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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 $m\ge0, n\ge1$, ...

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237 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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120 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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299 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 ...

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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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389 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

531 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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349 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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112 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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261 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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285 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 ...

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186 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.
...

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221 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 ...

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364 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 ...

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151 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
...

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459 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 ...

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146 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 ...

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

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137 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
$$
...

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257 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 ...

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### 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 ...

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164 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 ...

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493 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)} ...

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401 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}} = ...

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79 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 ...

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498 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 ...

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288 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 $0$?

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114 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 ...

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142 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 ...

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997 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 ...

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791 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 ...

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311 views

### Series with iterated log's: does it converge?

This came up in our office today. Let
$$f(x) = \begin{cases} x & \mbox{if } x\leq 1 \cr x\cdot f(\ln(x)) & \mbox{otherwise}\end{cases}$$
Does this series converge?
$$ \sum_{n=1}^\infty ...

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300 views

### Computing the limit of a certain recursively defined sequence

The following is not exactly a research question (it was originated from manufacturing of exercises for calculus), and has no other motivation than explaining a phenomenon. I apologize if it is ...

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131 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 ...

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1k views

### The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...

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191 views

### Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...

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180 views

### On methods for dealing with recursively defined sequences

Define $a_1=8$ and $a_n=\frac{4^{n+1}-2^{n+2}\sqrt{4^n-a_{n-1}}}{2}$ for $n\geq 2$.
By means of harmonic analysis methods it can be shown that $a_n$ converges to $\pi^2$ (this being the first ...

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234 views

### sequences - recurrence relation [closed]

I have to find the expression of $(y_n)$ defined by :
$$y_{n+1}=a y_n+b z_n+c$$
where $(z_n)$ is an arithmetico-geometric sequence :
$$z_{n+1}=d z_n+e$$
and $a,b,c,d,e$ real numbers.
Thank you ...

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226 views

### Convergence of a function in a metric space to its metric.

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric:
If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...

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**4**answers

808 views

### Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...

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664 views

### What is the theoretical interest of finding closed-form sols. of infinite series?

Hi,
I was reading this when I came across Gourevitch's conjecture.
My understanding is that solutions to these series are of practical interest. If one encounters such a series, being able to solve ...