Tagged Questions

4
votes
1answer
42 views

Chain rule proof

Let $a \in E \subset R^n, E \mbox{ open}, f: E \to R^m, f(E) \subset U \subset R^m, U \mbox{ open}, g: U \to R^l, F:= g \circ f.$ If $f$ is differentiable in $a$ and $g$ differentiable in ...
1
vote
1answer
17 views

Asymptotic Approximation and Sign Convention

When I write the asymptotic approximation of a function, does the sign convention matter? i.e. suppose I have (though the formula might not make sense) $$f_n(x)=x^2+\dots-O(n),$$ If my function is ...
14
votes
2answers
469 views

Asymptotics of sum of binomials

How can you compute the asymptotics of $$S=n + m - \sum_{k=1}^{n} k^{k-1} \binom{n}{k} \frac{(n-k)^{n+m-k}}{n^{n+m-1}}\;?$$ We have that $n \geq m$ and $n,m \geq 1$. A simple application of ...
1
vote
4answers
91 views

$\lim_{x\rightarrow\infty}\sin(x)$?

In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function. Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
1
vote
0answers
49 views

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 ...
2
votes
0answers
103 views

A proof of Stirling's Formula

I need to gain understanding of a proof of Stirling's formula. I have read through Tim Gowers' and Terence Tao's but I'm struggling to follow them. How rigorous is this proof, if at all? Thank you. ...
0
votes
1answer
80 views

Working with the ~ (tilde) notation (asymptotic analysis)

For positive functions $f$ and $g$ on real domains, define $f(n) \sim g(n)$ to mean $\displaystyle\lim_{n\to\infty}\frac {f(n)}{g(n)}=1$. Given that $$\frac{n^{n+\frac12}}{e^{n-1}n!}\sim\frac ...
5
votes
2answers
99 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
3
votes
2answers
66 views

Asymptotic rate of growth of a sum

Consider $$\Phi_0(x) = \sum_{i=0}^{\infty} (1-x)^i,$$ where $x \in (0,1)$. As $x \rightarrow 0$, $\Phi_0(x)$ blows up as $\Theta(1/x)$. Similarly, consider $$ \Phi_1(x) = \sum_{i=0}^{\infty} i ...
11
votes
1answer
146 views

If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?

If $\lambda_n,\mu_n \in \mathbb{R}$, $\lambda_n \sim \mu_n$ as $n \to +\infty$, and $\mu_n \to +\infty$ as $n \to +\infty$, is it true that $$ \sum_{n=1}^{\infty} \exp(-\lambda_n x) \sim ...
2
votes
0answers
109 views

Proof that limit goes to zero without Riemann-Lebesgue lemma

Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval). I know that the following $$ \lim_{\epsilon \to 0_+} ...
1
vote
1answer
110 views

Orders of Growth between Polynomial and Exponential

What is known in contemporary mathematics about orders of growth for functions that exceed any degree polynomial, but fall short of exponential? This is a subject for which I've found little ...
6
votes
1answer
96 views

Landau Notation Properties

I would like to take the limit $x\to 0$ and prove or disprove the following statements concerning Landau notation. (a) $O(x^{3/2})\subset o(x)\subset O(x^{1/2})$ (b) $(1+O(x))^2=1+O(x^2)$ I know ...
0
votes
2answers
69 views

Wby can't this recurrence be solved by direct guessing?

I'm reading Introduction to Algorithms by Cormen et. al. and in a part there, they say that this recurrence : T(n)=T(n/2) + T(n/2) + 1 can't be proved by the ...
0
votes
1answer
68 views

asymptotic sequence

I am asked to prove that $x^n(a+\cos(x^{-n})$ is an asymptotic sequence for $n=0,1,2,...$, $a>1$, $x\rightarrow 0$ but its derivative wrt x isn't an asymptotic sequence. ...
0
votes
1answer
78 views

An argument to prove asymptotic expansions

I have a real number $I_h$ depending on a small parameter $h>0$. I want to show that it has an asymptotic expansion in integer powers $h$, i.e. there exists a sequence $(J_k)_{k}$ such that $$ I_h ...
1
vote
1answer
62 views

Complete expansion of Laplace integral

Let $\varphi \in C^\infty (\mathbb R^n ;\mathbb R)$ such that 1) $\varphi(0)=0$ 2) $\varphi(x)>0$ on $\mathbb R^n\setminus 0$ 3) $\text{Hess}_{\varphi}(0)>0 $ and let $B_1(0)$ be the ...
11
votes
2answers
360 views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
2
votes
1answer
322 views

properties of a real analytic function

If there are a radius $r>0$ and constants $M,C\in\mathbb R$ for all $y\in U$ with $$|\partial^if(x)|\leq M\cdot i!\cdot C^{|i|}\space\space\space\space \forall x\in\mathbb B_r(y),i\in\mathbb ...
1
vote
1answer
373 views

Big O Notation question

I am trying to understand the Big-O and little-O notation, so I plotted 2 graphs which I have posted below, but I still dont really get the concept of it. What exactly does the ...
0
votes
2answers
606 views

big o notation / asymptotic for factorial

I want to write $g(x)=x!\cdot(x^4-1)$ in the big O notation $g\in \mathrm O(???)$ for $x\rightarrow\infty$. But I have no idea how to do this. Thanks for helping!
0
votes
0answers
205 views

Asymptotic equivalence?

Let there be two functions $f(x)$ and $g(x)$. If we consider $\lim_{x \rightarrow x_{0}} \frac{f(x)}{g(x)} = k$, we say that $k=1$, then $f(x)\sim g(x)$, $f(x)$ is equivalent to $g(x)$ as $x ...
3
votes
1answer
99 views

Dilogarithm asymptotics for an exponential parameter.

So this question is about this dilogarithm function. Assume the argument $z$ is real then I want to show the formula $$\operatorname{Li}_2(e^{-z})=\frac{\pi^2}{6} + z\log z -z+O(z^2) $$ as $z$ ...
2
votes
3answers
204 views

Intermediate growth rates

Is there any simple function/formula $f(n)$, which eventually dominates every $cn$ for every $c$, and is eventually dominated by $a \cdot n \cdot \ln^k(n)$ for every $a,k \in \mathbb{Z}$, where ...
3
votes
1answer
186 views

Evaluating a limit of the truncated exponential series motivated by the prime number theorem for $k$ distinct prime factors.

If $\pi_k(n)$ is the cardinality of numbers with k factors (repetitions included) less than or equal n, the generalized Prime Number Theorem is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln ...
4
votes
1answer
93 views

Deriving an asymptotic formula

I'm doing some exercises in a book on asymptotic analysis. While I think I found a solution to this problem, I'm not entirely sure if it's correct, and I want to make sure that I know what's going ...
6
votes
2answers
367 views

Equivalence to the prime number theorem

I was just reading this question and answer: How will this equation imply PNT and it raised a whole new question: Given that $\sum_{n\le x} \Lambda(n)=x+o(x)$, prove that $$\sum_{n\le x} ...
8
votes
1answer
174 views

Analytic number theory primer — sequences and series

For a book like Titchmarsh, or Iwaniec and Kowalski, it seems a thorough knowledge of asymptotics is a prerequisite. What are good books for training oneself in such manipulation of asymptotics, ...
5
votes
1answer
129 views

An estimate of a series

Suppose $s$ is not an integer, let $\lambda(s)=\min_{n≥0}|s+n|$. Show that $\sum\limits_{n=1}^{\infty}(\frac{1}{n+s}-\frac{1}{n})\ll\frac{1}{\lambda(s)}+\log(|s|+2)$.
2
votes
0answers
57 views

the nth power Logarithmic Integral [duplicate]

Possible Duplicate: Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$ So I want to show that $$\int_2^x \frac{1}{\log^n(t)}\mathrm dt=O\left(\frac{x}{\log^n(x)}\right)$$, ...
4
votes
1answer
118 views

How to estimate this integral

How to estimate the following integral: $$\int_e^x \log{\log{t}} dt$$ so that the error term is within $$O\left(\frac{x}{\log^2{x}}\right)$$. Assume $$x>e$$ Any hint?
10
votes
1answer
267 views

Asymptotic estimate for Riemann-Lebesgue Lemma

Let $f$ be a real-valued, $L^1$ integrable function on the interval $[a,b]$. Then the Riemann-Lebesgue Lemma tells us that: $$\int_a^bf(x)\sin(2\pi nx)dx\rightarrow0 \text{ as } ...
3
votes
4answers
130 views

Equality of outcomes in two Poisson events

I have a Poisson process with a fixed (large) $\lambda$. If I run the process twice, what is the probability that the two runs have the same outcome? That is, how can I approximate ...
6
votes
1answer
226 views

Algorithmic Analysis Simplified under Big O

Hi I am revising for my exams and I have the following inhomogeneous first order recurrence relation defined as follows: f(0) = 2 f(n) = 6f(n-1) - 5, n > 0 I ...
8
votes
2answers
334 views

How does Lambert's W behave near ∞?

How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of $$\frac{W(x)}{\ln(x)}$$ near $\infty$ (but along the positive real line, if that ...
10
votes
2answers
299 views

Positive integers $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}} > 1$ satisfying $\sum_{i = 1}^{n} p_{i}^{-r_{i}} < 1$

A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is unitary if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or ...
2
votes
0answers
153 views

Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log ...
10
votes
5answers
467 views

Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$

If $x \geq 2$, then how do we prove that $$\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}} = O\Bigl(\frac{x}{\log^{n}{x}}\Bigr)?$$
3
votes
2answers
197 views

Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$

Are there known sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (The factor of +1 in the ...
6
votes
3answers
454 views

Numerically estimate the limit of a function

Is there an algorithm that will allow me to numerically compute the limit of a function f(x) in a principled way? The most naive algorithm would be to continue to compute the function for larger ...
2
votes
1answer
295 views

Limit of the sequence $nx_{n}$ where $x_{n+1} = \log (1 +x_{n})$

Suppose $x_{1}>0$, and consider the sequence, $\{x_{n}\}$ defined as follows: $$x_{n+1}=\log(1+x_{n}) \quad n\geq 1 $$ Find the value of $\displaystyle \lim_{n \to \infty} nx_{n}$ I am having trouble ...