How can to find asymptotic form of power series?

The Power series is $ $ R_n(x)=\Sigma_{p=2}^{n+1}\frac{(p-1)(2n-p)!}{n!(n+1-p)!}x^p$ $

It says the following:

For large $ n$ , the asymptotic form of $ R_n(1/\beta)$ can be obtained by looking at the values of $ p$ which dominate the sum

  • $ p$ of order $ 1$ for $ \beta > 1/2$ ,
  • $ p$ of order $ \sqrt n$ for $ \beta = 1/2$ , and
  • $ p \propto (1 – 2\beta)/(1-\beta)$ for $ \beta < i$

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Asymptotic Bound on Minimum Epsilon Cover of Arbitrary Manifolds

Let $ M \subset \mathbb{R}^d$ be a compact smooth $ k$ -dimensional manifold embedded in $ \mathbb{R}^d$ . Let $ \mathcal{N}(\epsilon)$ denote the size of the minimum $ \epsilon$ cover $ P$ of $ M$ ; that is for every point $ x \in M$ there exists a $ p \in P$ such that $ \| x – p\|_{2}$ .

Is it the case that $ \mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$ ? If so, is there a reference?

Always exists an asymptotic component with large size in a minimal subshift?

Let $ (X, T)$ be a minimal subshift, i.e. $ X$ is a closed $ T$ -invariant subset of $ A^\mathbb{Z}$ , where $ T$ is the shift. A pair $ x,y\in X$ is asymptotic if $ d(T^nx, T^ny)$ goes to zero as $ n\to\infty$ . Always exists such a pair when $ X$ is infinite: for every $ n\geq1$ there exists $ x^{(n)}, y^{(n)} \in X$ such that $ x^{(n)}_0\not= y^{(n)}_0$ and $ x^{(n)}_{[1,N]}= y^{(n)}_{[1,N]}$ (if not, for some $ n$ , $ x_{[1,n]} = y_{[1,n]}$ implies $ x_0=y_0$ , i.e., $ x_{[1,n]}$ determines $ x_0$ , and this forces $ X$ to be periodic), and any pair of convergent subsequences of $ x^{(n)}$ and $ y^{(n)}$ will do the trick.

My question is: do exist $ k$ -tuples of asymptotic points, for every $ k\geq1$ ? More precisely, is it true that for every $ k\geq1$ there exists $ x_1,\dots,x_k\in X$ such that $ $ \lim_{n\to\infty}d(T^nx_i, T^nx_j) = 0\ \forall i,j$ $

Asymptotic analysis with factorial and exponential

I’m solving a complexity question where I have:

$ $ n!/2^n $ $

The goal is to find an upper bound for this.

My idea is using the fact that: $ $ n! = O(n^n)$ $ $ $ n!/2^n = O((n/2)^n) = O(n^n)$ $

But is a correct upper bound, and if so is it the tightest upper bound that can be found? I know that n! is asymptotically larger than 2^n, but I’m struggling to do a tighter analysis.

Minimization with asymptotic assumption

Given the function

$ g(n,m)=\min\Big\{f(a,b)+f(n-a,c)+f(n,m-bc)\Big|\a,b,c\ \ \text{with} \left\{\begin{matrix} a,\ b,\ n-a,\ c,\ m-bc \geq 0 \ b\leq a! \ c\leq (n-a)! \ \end{matrix}\right. \Big\} $

Assuming that $ n,m\geq 0,\ ((\lceil n/2\rceil)!)^2\leq m\leq n!,\ f(n,m)=\Omega (n)$ ,

is it true that $ g(n,m) \geq 2f(\lfloor n/2\rfloor ,(\lfloor n/2\rfloor)!)+f(n,m-((\lceil n/2\rceil)!)^2)$ ?

I tried KKT conditions, but can’t derive this (as it contains factorial).

Also, it seems that the condition $ f(n,m)=\Omega (n)$ implies that $ f$ is convex on our domain (and thus, satisfies the regularity condition for using KKT), but I managed to prove it only if $ f$ is polynomial.

So I am fully stuck in this…

Any help would be highly appreciated!

conditions for asymptotic comparison to hold

I have the following simple dynamical system: \begin{align} x_1′ &= a – f(x_2)x_1\ x_2′ &= bx_1 – cx_2, \end{align} where all parameters and initial conditions are positive. $ f(x_2)$ is a positive and increasing function with respect to $ x_2$ . Suppose I want to study the asymptotic behavior of this system and decide to do the following.

First, I note that the more $ x_1$ I have, the more higher the production rate for $ x_2$ will be. Secondly, I note that the larger $ x_2$ , the higher $ f(x_2)$ would be. Thus I choose to replace $ f(x_2)$ by a function $ g(x_1)$ such that:

  1. $ g(x_1) > 0$ and $ \frac{dg}{x_1} > 0$ .

  2. $ g(x_1)$ gives the same fixed points for $ x_1$ (perhaps through something like a quasi-steady-state-approximation for $ x_2$ ).

Together, I obtain: \begin{equation} x_1′ = a – g(x_1)x_1. \end{equation}

Due to the construction, the asymptotic behavior of $ x_1$ in this equation should be the same as that of $ x_1$ in the original equation. I tried this out with $ f$ and $ g$ being simple hill equation and it works.

This is just a toy example. My question is: for higher dimension and more complicated functional responses, if I only care about asymptotic behavior, when can something similar can be carried out? I would appreciate any references on this topic.