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

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### Usage of multinomial theorem with infinite series

What are the conditions for using the multinomial theorem with infinite series? I have an expression but I don't know if I can use it. The expression is:
$$
\left[\sum_{m=0}^{\infty} \frac{\mu^m ...

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**3**answers

387 views

### Summation of a series

I would like to sum the series
$$
\sum_{n=0}^\infty \frac{1}{(1+a^2 (n+1/2)^2) ^{3/2}} .
$$
It arose when trying to perform a calculation on superconductivity. In particular I am interested in its ...

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votes

**1**answer

113 views

### Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define
$$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...

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votes

**1**answer

567 views

### Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series
$$f_{\sigma}(z) = ...

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**2**answers

119 views

### series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...

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**2**answers

334 views

### Asymptotic behaviour of sequence

I am interested in the sequence
$$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$
where $p(n)$ is a polynomial equation.
When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...

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votes

**1**answer

320 views

### Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series
$$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$
where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...

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**2**answers

267 views

### norm of the matrix series

The goal is to obtain an upper bound for the norm of the vector
$$
\left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\|
$$
for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...

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**1**answer

767 views

### Can an infinite number of mathematicians guess the number in a box with only one error?

In this question the following observation was made:
Consider a sequence of boxes numbered 0, 1, ... each containing one real number. The real number cannot be seen unless the box is opened.
Define ...

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**1**answer

170 views

### Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces

In this Math Stack Exchange post, I proved the following result.
Theorem: Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x_{n})_{n \in \mathbb{N}} $ is ...

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296 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}}} + ...

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**3**answers

1k views

### Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question):
The Riddle:
We assume ...

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**2**answers

171 views

### Finding the min of a sequence related with factorials

Let $N,n$ be natural numbers.
Let us define $a_n=m$ when $N!$ can be divided by $(n!)^m$ and it cannot be divided by $(n!)^{m+1}$.
For a given $N(\ge 2)$, let $\min(N)$ be the min of $na_n\ (2\le ...

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111 views

### Multivariate generating function

I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^ {n-k} q^k$ where $p$ and $q$ are distinct primes ...

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**1**answer

654 views

### Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum?
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{p<n}_{p\ \ prime} \frac{1}{\log{n}} ...

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**1**answer

151 views

### Infinite series - analytical solution

Analytical Solution is required for:
$$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$
$$\sum_{n=0}^\infty ...

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**1**answer

350 views

### Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$

I am currently interested in the following sequence:
$$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...

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63 views

### Proof that Newton expansion over derivatives has the properties of an integral [duplicate]

Let's consider a Newton expansion over consecutive derivatives of a function:
$$F(x)=\sum_{m=0}^{\infty} \binom {-1}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
Can it be proven that such ...

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**1**answer

413 views

### How to find the coefficients of a poor-converging series?

I have the series
$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$
and the boundary conditions
$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...

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92 views

### A generalization of alternating series involving modulus?

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

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**1**answer

418 views

### Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...

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125 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 ...

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95 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 ...

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615 views

### $\sum_{n=1}^{\infty}\frac{1}{a_n}=\infty$ $\sum_{n=1}^{\infty}\frac{1}{b_n}=\infty$ but $\sum_{n=1}^{\infty}\frac{1}{a_n+b_n}=c, c\in R$ [closed]

The following question is inspired from: Defining the slowest divergent series.
Let $a_n$ and $b_n$ be two strictly increasing sequences of natural numbers,with ...

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179 views

### Transcendental numbers as infinite products of sides of squares

We can obtain such an infinity of sides of squares by continuously increasing the length of the side of the inscribed square to the length of the circumscribed square of a circle with diameter equal ...

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**1**answer

136 views

### 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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869 views

### 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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497 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

446 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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**0**answers

206 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

79 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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115 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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342 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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469 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

492 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

249 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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189 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

203 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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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

227 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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**3**answers

1k views

### 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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**1**answer

252 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

230 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

425 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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486 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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**3**answers

583 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.
...

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**2**answers

737 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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**1**answer

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 ...