Questions tagged [asymptotics]

Question 1

Q1. $$\int \frac{3x^{2} + 4x - 1}{(x^{2} + 1)^{2}\sqrt{x+1}}\, dx$$ $\textbf{A. }\frac{\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{B. }-\frac{2\sqrt{x+1}}{x^{2}+1} + C$ $\textbf{C. }-\frac{x}{(x^{2}+1)\sqrt{x+...

Question 2

I recently came across a surprising (to me) limit that I wanted to try to understand a bit better. The limit is: $$ \lim_{N \to \infty} | N \cdot ( \ln(2) - (1 - 1/2 + 1/3 + \ldots + (-1)^{N-1} / N)| ...

Question 3

Let $Q(x)$ be the number of square-free integers up to $x$. The asymptotic formula is well known, namely $$Q(x) = \frac{x}{\zeta(2)} + \Delta(x)$$ with error term $\Delta(x)$. It is also known that $\...

Question 4

Suppose I have the recurrence relation $$ f(r) = \alpha + \beta \sum_{s=1}^{r-1}f(s), \qquad r \geq 2$$ for some $\alpha, \beta > 0$ and initial condition $f(1) = \gamma > 0$. This gives a ...

Question 5

As in https://en.wikipedia.org/wiki/Ackermann_function, the Ackermann Function is defined by $$A(0,n) = n+1\\ A(k+1,0)=A(k,1)\\ A(k+1,n+1)=A(k,A(k+1,n)). $$ The inverse Ackermann Function is defined ...

Question 6

It's my understanding that $\frac{1}{2^{1/x}-1}=O(x)$ (as $x$ approaches $+\infty$), but I'm struggling to prove why this is the case. How does one prove this? Thanks!

Question 7

Question Find an asymptotic formula for the integral $$I_n = \int_0^1 \frac{1-(1-t)^n}{t} dt$$ as $n\to \infty$. source: this question (the case $b=2$). I thought about Laplace's method but the ...

Question 8

Equations of the form $$y = \sin \left(\frac{y^s}{t} - k \right)$$ are surprisingly well approximated as $$y = - \sin k$$ for a large range of $s$ $t$ and $k$ values: Why? I understand why this is so ...

Question 9

We have $\left\lfloor x\right\rfloor$ as the floor function and $\left\{x \right\}$ as the fractional part. Looking for the asymptotic expansion as $x \rightarrow \infty$ of $$DW (x)=\sum_{n \le x} \...

Question 10

I've been strugling to find an asymptotic sequence depending on both $\{ap\}$ and $a \in (0,1/2) $ of the following sum defined for all positive integers $p$ : $$ \sum_{k = 0}^{\lfloor ap \rfloor} \...

Question 11

Let $\mathcal{F}$ be the space of all increasing continuous functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ s.t. $\lim\limits_{x\to\infty}f(x)=\infty$. Consider the equivalence relation $$ f \sim g\iff ...

Question 12

Suppose we have an interval say $D\to D+q$. Let $N(D,q)$ be the number of integer triplets $(a,b,c)\in\mathbb{Z}^3$ such that $D\le \sqrt{a^2+b^2+c^2}\le D+q$. While doing physics, we very easily ...

Question 13

On pages 291-294 of Bender & Orszag (Advanced Mathematical Methods for Scientists and Engineers-Asymptotic Methods and Perturbation Theory) they develop the full asymptotic expansion of $J_0(x)$ ...

Question 14

When thinking about an unrelated problem, I wanted to sum $$\sum_{k\in[\sqrt n,2\sqrt n]}\#\{j\leq k/5:\gcd(k,j)=1\}$$ and I wanted this sum to be $\Theta(n)$. From the wikipedia I can see that if we ...

Question 15

I have this series $$\sum_{n=1}^{\infty} \left[2^{\left(n^{x}-n^{x} \cos \frac{1}{n^{2}}\right)}-1\right].$$ let $a_n$ the general term of the series, $$ a_n = 2^{\left(n^x - n^x \cos \frac{1}{n^2}\...

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Questions tagged [asymptotics]

How are these integral behaviour approximations different

Why do the Genocchi numbers appear in this series?

Prove that the number of sign changes $\sim x^{3/4}$ (square-free integers)

Perturbation of linear recurrence

Two definition of Inverse Ackermann Function

Determining the asymptotic growth of a function [closed]

Asymptotics of the integral $\int_0^1 \frac{1-(1-t)^n}{t} dt$

Why is $y = \sin \left(\frac{y^s}{t} - k \right)$ outstandingly approximated by $y = - \sin k$, even for large $k$?

Asymptotic expansion as $x \rightarrow \infty$ of $\sum_{n \le x} \lfloor\frac{x}{n}\rfloor \left\{{2\sqrt{\lfloor\frac{x}{n}\rfloor}}\right\}$

Asymptotic equivalence of a binomial Sum :

Topology on the space of asymptotic growth rates

Proof/counterproof of a statement which seems obvious [closed]

Asymptotic Expansion of Bessel Function using Sommerfeld Contour

Restricted Euler function

Divergence, convergence of the series $\sum_{n=1}^{\infty} \left[ 2^{(n^x - n^x \cos(1/n^2))} - 1 \right]$

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