4
votes
2answers
85 views
+50

An asymptotic expression of sum of powers of binomial coefficients.

Let $k$ be a fixed positive number and $n$ an integer increasing to infinity. Then $$\sum_{\nu =0}^n \binom{n}{\nu}^k \sim \frac{2^{kn}}{\sqrt{k}} \left( \frac{2}{\pi n} \right)^{\frac{k-1}{2}}.$$ ...
2
votes
1answer
41 views

terminology relating to o(1)

If someone says, for example, "I have an algorithm that runs in time $n^2+\varepsilon$ for any constant $\varepsilon>0$", the interpretation for this statement seems to be that for any constant ...
0
votes
3answers
67 views

Prove that $1^{k} + 2^{k} + \cdots + n^{k}$ is $O (n^{k+1})$

I have the following to prove: $1^{k} + 2^{k} + \cdots + n^{k} \text{ is }O (n^{k+1})$ I have done the following: $$\frac {1^{k} + 2^{k} + \cdots + n^{k}}{n^k} \leq n$$ Am I on the right track? I ...
3
votes
1answer
27 views

If $f$ is equal to an affine function up to $1$-th order at $a$, then $f$ is differentiable at $a$, proof more subtle then it appears?

I came across the following exercise: Two functions $f, g : \mathbb R \to \mathbb R$ are equal up to $n$th order at $a$ if $$ \lim_{h \to 0} \frac{f(a + h) - g(a + h)}{h^n} = 0. $$ Show that $f$ ...
1
vote
1answer
33 views

Clarification: how to get the following asymptotics

I'm having some trouble justifying some steps in a paper. Let $a_n$ be an increasing sequence of integers satisfying $n! \le a_n \le 2(n!)$, and let $f:\mathbb{N} \to \mathbb{N}$ be a function ...
1
vote
1answer
21 views

Relationship between big O notation and exponential type

Let $f: \mathbb{R} \to \mathbb{R}$, $C\in \mathbb{R}$. What, if any, is the difference between "$ f = O(e^{Cx}) $" and "$f$ is of exponential type $C$"? If they're different, is it possible to ...
0
votes
1answer
20 views

Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
0
votes
1answer
29 views

Big-O Analysis: Max Bounded by the Sum

I have been asked to show that: $$ \mathcal{O}(Max\{ f(n), g(n) \}) = \mathcal{O}(f(n) + g(n)) $$ I have seen explanations of similar problems, but this is the first time I have encountered the ...
0
votes
1answer
36 views

Big O notation for complex-valued functions of a real variable

Let $f,g:\mathbb R\to\mathbb C$. Is there a standard notion of $f = O(g)$? If I had to take a stab at a definition, I'd try something like $f = O(g)$ provided there exists $M>0$ and ...
3
votes
2answers
62 views

if $f(x) \sim g(x)$ is $ \sum f(k) \sim \sum g(k)$

if $f(x) \sim g(x)$ as $x \to \infty$ then is $\sum_{k=1}^N f(k) \sim \sum_{k=1}^N g(k)$ as $N \to \infty$? Intuitively, i should think so because as $k$ gets larger $f$ and $g$ get closer so it ...
5
votes
3answers
124 views

Proof that $J_{\nu}(x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu \rightarrow \infty$

I'm working through the exercises of Bender and Orszag's famous book, but I got stuck in 6.25 (a), in which it is asked to prove that $$J_\nu (x) \sim (x/2)^\nu / \Gamma(\nu+1) \; \text{as} \; \nu ...
0
votes
1answer
10 views

A question on an asymptotic combinatorial expasion

Suppose we are given $(\lambda a + \bar{\lambda}b+O(\lambda^2))^{n}$, where $0 < \lambda < 1$ and $\bar{\lambda} := 1-\lambda$; also, $0 < a,b < 1$. $O(\cdot)$ is the traditional Big-Oh ...
4
votes
2answers
126 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
0
votes
1answer
31 views

Is it possible to integrate this asymptotic expansion?

Suppose that some smooth function $f \in C^\infty\bigl(\mathbb R^n \times (0,+\infty)\bigr)$ possesses an asymptotic development $$ f(x,t) \sim t^{-\alpha} e^{ith(x)} \sum\limits_{k=0}^{+\infty} ...
3
votes
1answer
188 views

Asymptotic expansion of $\int_0^{2\pi}ae^{x\cos a}da$

I want to find the first two leading terms of the expansion of $\int_0^{2\pi}ae^{x\cos a}da$ Well in $[0,2\pi]$ $\cos a$ has has maxima $0,2\pi$ so I rewrite the integral to ...
2
votes
1answer
75 views

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 ...
1
vote
1answer
239 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 ...
0
votes
0answers
46 views

How to check if a function is negligible?

Let $\epsilon(x)$ be a negligible function. Let $p$ be a polynomial such that $p(k) \geq 0$ for all $k > 0$. What can we say about $\epsilon(p(k))$? Is this a negligible function? If yes, ...
3
votes
0answers
43 views

Growth rate of $\exp(log^{a}(x))$ slower then any power of $x$.

So I'm trying to show that for $0<a<1$ and for $\epsilon >0$ that $\exp((\log x)^{a})=\mathcal{O}(x^{\epsilon}).$ So this amounts to showing that ...
0
votes
1answer
49 views

a question concerning asymptotics

I have a rather simple question I need an answer to that I have been unable to answer and was wondering if anyone knew any results that pertain to this. It's very simple to state and I believe the ...
3
votes
1answer
55 views

The statements $f(n) = O(n^{\epsilon})$ for all $\epsilon > 0$ and $f(n) = n^{o(1)}$.

Consider the statements \begin{align} \tag{A} f(n) &= O(n^{\epsilon}) \text{ for all } \epsilon > 0 \\ \tag{B} f(n) &= n^{o(1)} \end{align} Questions: It's clear that (B) implies (A). ...
3
votes
1answer
83 views

Cesaro means and equivalent sequences

Let $(u_n)$ be a sequence of complex numbers that converges in mean (Cesaro convergence). Let $(v_n)$ be a sequence such that $v_n\sim u_n$. Does the sequence $(v_n)$ converge in mean? Here is ...
13
votes
5answers
920 views

Is there any nonconstant function that grows (at infinity) slower than all iterations of the (natural) logarithm?

Is there any nonconstant function that grows (at $\infty$) slower than all iterations of the (natural) logarithm? Thanks for your help.
0
votes
1answer
43 views

Does $ \log(x)^{x^a}$ eventually dominate $x^k$?

Does $ \log(x)^{x^a}$ eventually dominate $x^k$ for all $a\gt 0$ and for all positive integers $k$? And if so, how does one prove this? Thanks a lot for your help.
3
votes
2answers
141 views

Show $S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}$ is $O(t^p)$ at zero

An old qualifying exam problem: For $t>0$, define $$S(t) =\sum_{n=-\infty}^\infty\sin{(n^2t^2)}e^{-tn^2}.$$ Show that $S(t) = C t^p + o(t^p)$ as $t\to 0$ . Find $C$ and $p$. There are a couple of ...
0
votes
1answer
78 views

Derive asymptotic behavior of inverse of the normal cdf with respect to 2^n

I have a normal distribution $\mu = 0$ and $\sigma = 0.58n$ where $n > 0 $ and I am trying to derive the asymptotic behavior of the following equation: ...
1
vote
1answer
28 views

Asymptotic behaviour of a function of a bivariate normal vector

Let $(Z_1,Z_2)$ be a bivariate standard normal vector and $x\in\mathbb{R}$. We consider $$f(\sigma_l):=\left| \operatorname{E}[1\{Z_1\leq x/\sigma_l\}1\{Z_2\leq ...
0
votes
1answer
15 views

Asymptotics of a real sequence

Let $(a_n)_{n\in\mathbb{N}}$ be a real sequence with $a_n\in O(n^d)$ $(d\in (-1,0))$. Now we consider the expression $$ b_n:=(1-\sqrt{1-a_n}).$$ Is $b_n\in O(\sqrt{n^d})$? Thanks!
0
votes
1answer
31 views

Asymptotic behaviour of real sequences

Let's say we have two real sequences $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ with $c_n\in o(\frac1n)$ (i.e. $c_n(\frac1n)^{-1}\xrightarrow{n\rightarrow\infty}0$). And for all ...
2
votes
1answer
40 views

Would this be bounded

Let $a_{m}$ be supremum of the minimum of the angle between the line segments between any $m$ points, in which the supremum is taken over all configurations of $m$ points. Is $\sqrt{m}a_{m}$ bounded ...
3
votes
4answers
163 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
1
vote
0answers
61 views

function defined as an integral involving Bessel functions

i need to analyze a function of the form $$F(x,y) = \int_0^{1} e^{-(1+s)\alpha x}\sinh((1-s)\beta y) I_0(\sqrt{(x^2-y^2)s}) ds $$ Where $I_0$ is the modified Bessel function. $x>y$ always. ...
0
votes
1answer
170 views

Asymptotic notation meaning in transitive relation

I'm attempting to prove the transitive relation on $\theta$ and I'm having trouble understanding the meaning of one of the symbols used. Here is the transitive relation: $f(n) = \theta(g(n)) ...
1
vote
0answers
287 views

Relation between the exponential function and the modified bessel function of second kind

I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the ...
4
votes
3answers
120 views

Convergence of partial sums of real sequences

For all $i\in\mathbb{N}$, let $(a_{i,n})_{n\in\mathbb{N}}$ be a real sequence that tends to $0$ for $n\rightarrow\infty$. It holds also that $|a_{i,n}|\leq1$ for all $i,n\in\mathbb{N}$. Is it possible ...
2
votes
2answers
215 views

$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)$

$\sum _{p\leq n}\frac{\ln p}{p}=\ln n+O(1),n\geq 2,$ where $p$ is a prime number, prove: $$\sum _{p\leq n}\frac{1}{p}=C+\ln \ln n+O\left(\frac{1}{\ln n}\right)~~~(1)$$ one examination ...
4
votes
1answer
97 views

How can I approximate $\sum\limits_{k=4}^{\infty}\Pr(X=k)[{\Pr(X\le k)}^6 - {\Pr(X\le k-4)}^6]$ for $\lambda \to +\infty$?

$X$ is a Poisson random variable and the probability mass function is given by: $$\Pr(X = k) = e^{-\lambda}\frac{{\lambda}^k}{k!}$$ I’ve got a probability function $f(\lambda)$ $$f(\lambda) = ...
2
votes
3answers
70 views

Asymptotic behaviour of $1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)} \right) ^2$

I know that $$\lim_{n\rightarrow \infty}\frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} \Gamma(n)}=1,$$ but I'm interested in the exact behaviour of $$a_n =1- \left( \frac{\Gamma(n+\frac{1}{2})}{\sqrt{n} ...
4
votes
0answers
151 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
0
votes
2answers
84 views

Why is big-Oh multiplicative?

If $f$ is $O(g)$ over some base, this means that $f(x) = \beta(x)g(x)$, where $\beta$ is eventually bounded. So this means that eventually, $f$ is at most $c$ times $g$, where $c$ is some constant. ...
2
votes
1answer
57 views

Asymptotics at the origin of the convolution with an approximation to the identity.

In short, I am trying to find sufficient conditions for an approximation to the identity function $K_h$ so that, for $h$ small enough and fixed, the asymptotics at the origin of an $L^1 \cap L^2$ ...
1
vote
1answer
67 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
1
vote
0answers
41 views

Asymptotic growth over an interval

Given a function $f(x)$, we can define the new function $$ A_f(t) = \limsup\limits_{x\to\infty}\ (f(x+t) - f(x)) $$ Is there a place that this transformation has been studied? Also, given a positive ...
5
votes
1answer
138 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
25 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
548 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
162 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 $\cos(x)$ ...
2
votes
0answers
109 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
207 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
733 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 ...