3
votes
0answers
49 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
0
votes
1answer
147 views

Proving functions to be Big Oh

How do I determine if there exists a function $f$, such that \begin{equation} f(n) = {\mathcal O}(\log n), \end{equation} but \begin{equation} 2^{f(n)} ≠ {\mathcal O}(n). \end{equation} Is ...
3
votes
2answers
74 views

Why is $3^n$ not in $\Theta(2^n)$

How is it that $3^n$ not in $\Theta(2^n)$, while $log_3 n$ is in $\Theta(log_2 n)$ ?
0
votes
2answers
28 views

limit with exponential

I am trying to solve asymptotic relation between 2 functions: $$f(n)=2^n*n$$ $$g(n)=\frac {3^n}{n^2} $$ I started to solve $$\lim_{x\to \infty} \frac{2^n*n^3}{3^n}=\lim_{x\to \infty} (\frac ...
1
vote
1answer
58 views

Is there an “interesting” function that grows faster than $n^{kn}$ but slower than $2^{2^n}$ — relates to understanding googolplex

Motivation: I'm looking for some sort of convenient fact I can use to grasp the size of a googolplex. For a googol we observe a convenient one; it's very nearly equal to 70!. But for a googolplex I ...
1
vote
0answers
54 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
1
vote
1answer
352 views

Rate of growth of exponential functions

I have difficulties about proving the following: Prove that exponential functions $a^n$ have different orders of growth for different values of base $a>0$. It looks obvious that when $a=3$ it ...
2
votes
1answer
148 views

Rate of convergence of $\left[1+\frac{a}{x}\right]^x$ to $\operatorname{exp}[a]$ as $x\rightarrow\infty$

It's well-known that $\lim_{x\rightarrow\infty}\left[1+\frac{a}{x}\right]^x=\operatorname{exp}[a]$. I am wondering how fast does the limit converge as $x$ increases, and how the speed of convergence ...
3
votes
1answer
336 views

Comparing the asymptotic growth of two exponential functions

I'd like to compare the asymptotic growth rates of two functions: Cayley's formula for the number of trees on $n$ vertices: $n^{n-2}$ The number of possible graphs on $n$ vertices: $2^{n \choose 2} ...
3
votes
1answer
225 views

“Proof” that if f(x) ~x, e^f(x) ~ e^x

While it is not true that $f(x)\sim x \implies e^{f(x)}\sim e^x,$ I can't spot the error in this "proof" by induction--or at least I can't articulate it well. Let $f(x)\sim x$ and $x > 1$ P(1): ...
-1
votes
1answer
166 views

How to find asymptotic entire functions?

I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...