Articles of asymptotics

Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?

Let us assume that $f(n)=2^{n+1}$, $g(n)=2^n$ be two functions. Now, use limit to find $O(f(n))$: $\lim_{n\to\infty} \dfrac{2^{n+1}}{2^n}=2$. This is not equal to infinity, so the limit exists, hence $2^{n+1}$ belongs to $O(2^n)$. Now, according to transitivity property: $$2^n=O(2^{n-1}), \;2^{n-1}=O(2^{n-2}),\;\ldots ,\;2^i=O(2^{i-1})$$ so using transitivity property we can write $2^{n+1}=O(2^{i-1})$. We can go on extending this so […]

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let’s say $p$ is even. I’m interested in the asymptotic behavior of this for $n$ and $p$ both large. This is my attempt to make a finite sum similar to the series in this problem that might […]

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be any function that satisfies $$\lim\limits_{n\to\infty} \left(e^{\frac1e} + \frac1n\right)^{\wedge\wedge}\left[(10 n)^{1/2} + n^{A(n)} + C+o(1)\right] – n = 0$$ where $C $ is a constant. Then $\lim\limits_{n\to\infty} A(n) = \frac1e $ Conjectured by […]

Number-theoretic asymptotic looks false but is true?

Question Let $p_r$ be the $r’th$ prime. Is it true that, $$\sum_{r=1}^\infty s^r \ln(p_r) \sim \frac{s}{(1-s)} $$ I know this looks bizarre but kindly consider the argument below. I’m also interested in knowing if this technique (of summing to simplify via asymptotics) is known? Argument I recently observed an interesting behavior of the following series: […]

Growth of $\Gamma(n+1,n)$ and $\operatorname{E}_{-n}(n)$

Quite often when I ask W|A to solve something it gives me an answer in terms of $\Gamma(n+1,n)$ or exponential integral $\operatorname{E}_{-n}(n)$. Looking up the definition of the incomplete gamma function and the exponential integral I can get a formal definition but I have no feeling for how these function behaves. What is the asymptotic […]

Why $2n \int ^\infty _n \frac{c}{x^2 \log(x)} \sim n \frac{C}{n \log(n)}$?

I want to understand why we have $$ 2n \int ^\infty _n \frac{c}{x^2 \log(x)} \operatorname*{\sim}_{n\to\infty} n \frac{C}{n \log(n)} $$ where $c$ is a normalizing constant. I am unable to understand how the integral is removed.

What is the asymptotic bound for this recursively defined sequence?

$f(0) = 3$ $f(1) = 3$ $f(n) = f(\lfloor n/2\rfloor)+f(\lfloor n/4\rfloor)+cn$ Intuitively it feels like O(n), meaning somewhat linear with steeper slope than c, but I have forgot enough math to not be able to prove it…

Solve $\epsilon x^3-x+1=0$

I’m trying to find the expansion for the roots of this equation. I’ve found one root as $x\sim 1+\epsilon $. Now considering the dominant balance I want to rescale so that $\epsilon x^3\sim O(x) \Rightarrow x=O(1/\sqrt\epsilon )$ Setting $x=y(1/\sqrt\epsilon )$ where $y=O(1)$ I get the new equation $$y^3-y+\sqrt\epsilon=0$$ Now I want to substitute in $y\sim […]

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at large times, the graph of $\sin$ oscillates very rapidly and so you can take it to be zero. When […]

Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$

I want to analyze the behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$, i.e I want to show it behaves like $e^x(\frac{2}{x}+\frac{2}{3x^2}+…)$ as $x\rightarrow \infty$ I started by looking the at the exponent $\cos\sqrt{t}$, taking the derivative and setting it $0$ gives me $0,\pi^2,2^2\pi^2,3^2\pi^2,…$ My guess is to use Lapalce Method now but I do not know how. How can […]