# Tagged Questions

**2**

votes

**2**answers

298 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 ...

**8**

votes

**1**answer

299 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 ...

**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 ...

**6**

votes

**1**answer

237 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 ...

**5**

votes

**1**answer

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 ...

**1**

vote

**1**answer

376 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)} ...

**1**

vote

**0**answers

131 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

**2**answers

289 views

### Equidecomposable graphs, unimodality and asymptotics

I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs
$$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad ...

**1**

vote

**2**answers

316 views

### Asymptotics of Fourier coefficients of power-type functions

I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions
$f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic ...

**2**

votes

**2**answers

313 views

### Inverting an asymptotic series

I have the first few terms of a series of the form,
$y(x)=\ln(x)+x+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$.
Knowing that the inverse $x(y)$ exists, I am looking for method to write x in terms of ...

**8**

votes

**2**answers

769 views

### What is the series expression for (1+1/x)^x about x = \infty?

This seems like it must have been addressed somewhere already, but I cannot find it in any standard series tables.
I have the equation:
$f(z) = \left(1 + \frac{1}{z}\right)^z$.
What is the general ...

**4**

votes

**0**answers

193 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

**2**answers

595 views

### Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello!
Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...

**0**

votes

**2**answers

740 views

### Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the ...

**7**

votes

**4**answers

1k views

### Determining the asymptotic behavior of a series

I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable ...

**23**

votes

**4**answers

1k views

### Asymptotic growth of a certain integer sequence

Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence A019568 defined as follows:
$a(n):=$ the smallest positive integer $k$ such
...

**11**

votes

**5**answers

837 views

### Asymptotics of a Bernoulli-number-like function

Tony Lezard asked me the following question which seemed like it should not be too hard but which I did not immediately see how to answer. Define $f(n,k)$ recursively by $f(1,k) = 1$ and
$$f(n,k) = ...