Tagged Questions
1
vote
1answer
73 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
62 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
136 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
156 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
137 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 $P$ ...
