Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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Asymptotics for the Alternating Mertens Function

Are there any tight bounds, or any nontrivial ones actually, known for, with the lack of a better name, the Alternating Mertens Function: $$ S(n) = \sum_{k=1}^{n} \left((-1)^k \mu\left(k ...
8
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2answers
71 views

Reworking $\sum_{n \leq x} \frac{1}{n^s}$, where $n$ is relatively prime to some fixed $k$

For a fixed integer $k \geq 1$ and real $s>0$ I want to rework the partial sums $$\sum_{\substack{ n \leq x \\ \text{gcd}(k,n) = 1 }} \frac{1}{n^s}$$ in such a way that I can find an ...
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2answers
35 views

Efficiently calculating the 'prime-power sum' of a number.

Let $n$ be a positive integer with prime factorization $p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m}$. Is there an 'efficient' way to calculate the sum $e_1+e_2+\cdots +e_m$? I could always run a brute ...
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0answers
21 views

Asymptotic behavior of the solution of a 2nd order linear ordinary differential equation

In studying the harmonic oscillator, we encounter the equation $$ f'' +(2E - x^2) f = 0$$ What is the asymptotic behaviour of the solution to this equation for a generic $E$? Any good book on ...
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2answers
56 views

Limit of $\sqrt[n]{(x+1)…(x+n)} - x$ as $x \to +\infty$

Let $n \in \mathbb{N}^{\ast}$. I want to determine the following limit : $$ \lim \limits_{x \to +\infty} \sqrt[n]{(x+1)\ldots(x+n)} - x.$$ Let $x = \frac{1}{t}$ with $t \to 0$. It is equivalent to ...
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0answers
20 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
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2answers
70 views

An equivalent for $\sum_{n=0}^{\infty} e^{-x\sqrt{n}}$ as $x$ tends to $0^+$

I would like to obtain an equivalent form for $$ f(x)=\sum_{n=0}^{\infty} e^{-x\sqrt{n}} $$ as $x \rightarrow 0^+$. I tried without success to "remove" the $\sqrt{\cdot}$ in the summand by summing ...
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2answers
37 views

Find the leading order uniform approximation when the conditions are not $0<x<1$

$$\epsilon y''+y'\sin x+y\sin 2x = 0$$ with boundary conditions $y(0)=\pi$ and $y(\pi)=0$ as $\epsilon \rightarrow 0$. I don't know how to find out where the boundary layer is? I thought initially it ...
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0answers
19 views

Find the leading order uniform approximation to the boundary value problem $\epsilon y''+y'\sin x+y\sin 2x = 0$? [duplicate]

$$\epsilon y''+y'\sin x+y\sin 2x = 0$$ with boundary conditions $y(0)=\pi$ and $y(\pi)=0$ as $\epsilon \rightarrow 0$. I don't know how to find out where the boundary layer is? I thought initially it ...
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1answer
24 views

A probably simple big $\mathcal{O}$ question

I have a probably simple big $\mathcal{O}$ question. Is the following statement correct? $$\mathcal{O}(x \log x)=\mathcal{O}(\sqrt x \log x)$$ why?
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1answer
43 views

power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
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1answer
32 views

How can I write in Landau notation (or the like) that $2^x/x$ rises almost as fast as $2^x$?

Since $2^x \not\in O(2^x/x)$, we do not have $O(2^x/x)=O(2^x)$. But since $x$ rises linearly and $2^x$ exponentially, $2^x/x$ rises almost as fast as $2^x$. Can I somehow express this in Landau ...
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1answer
37 views

Obtain the leading order uniform approximation of the solution to: $\epsilon y'' +(1+x)^2y'+y=0$?

Obtain the leading order uniform approximation of the solution to: $\epsilon y'' +(1+x)^2y'+y=0, y(0)=0 y(1)=1$ as $\epsilon \rightarrow 0$. I am completely lost. Am i right in doing this? Since ...
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0answers
44 views

$\epsilon y''+\sqrt{x}y'+y=0$, show there is no boundary layer at $x=1$ and a boundary layer of $\epsilon^{\frac{2}{3}}$ at $x=0$?

I'm so lost. If I use quadratic formula I obtain that: $$y(x) = ae^{-2\epsilon\sqrt{x}}+be^{-2x\sqrt{x}+2\epsilon\sqrt{x}}$$ with the boundary conditions $y(0)=0$ and $y(1)=1$ but how does this lead ...
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0answers
17 views

Do we recognize higher degree asymptotes beyond Horizontal and Oblique?

I am reading a textbook, and it talks about doing synthetic division in order to rewrite a function into the quotient $$R(x)=\frac{p(x)}{q(x)}= f(x) + \frac{r(x)}{q(x)}$$ Since $\frac{r(x)}{q(x)}$ ...
3
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1answer
45 views

Determining the asymptotics of the Summatory function of an Arithmetic Function

We define the arithmetic function: $\displaystyle f(n) = \max\limits_{p^{\alpha} || n} \alpha$, that is if $\displaystyle n = p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ (prime factorization of $n$) then ...
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1answer
21 views

Subtraction of functions with BigO

When trying to assess the BigO of two functions that are added together, we take the max of the two. What happens if there is subtraction instead of addiiton? for instance: $$f(n) = bigO(n^3) $$ $$ ...
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1answer
24 views

Strict upper and lower bounds of a sum (Big-Theta)

I am trying to find a function f(k) such that $S_k=\sum_{n=1}^{k^2-1}(\lfloor\sqrt{n}\rfloor)=\Theta(f(k))$. What I have done so far: Ignoring the floor asymptotically we get: ...
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2answers
21 views

Analytic Combinatorics to asymptotically estimate the number of objects of size at most n?

I have read some bits of Flajolet's and Sedgewick's book on Analytic Combinatorics. I am quiet curious as how to asymptotically estimate the number of objects of size at most n. Suppose for example ...
3
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2answers
56 views

Can one apply a WKB method to an inhomogeneous first order differential equation in order to find the asymptotic expansion of the solution?

Consider \begin{equation} \varepsilon \frac{dy}{dx} = Q(x)y + R(x) \end{equation} where $\varepsilon$ is a small parameter. Can one apply a WKB method to find an asymptotic expansion for the ...
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0answers
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asymptotic distribution (Inference statistic) [closed]

Assume $Pr(A_{j}) = p^{0}_{j}$, $j = 1,...,k$. Obtain the asymptotic distribution of the statistic $f(\hat{p_{1}},...,\hat{p_{n}})$ = $n\sum_{j=1}^{k}p^{0}_{j}(1 -\frac{\hat{p_{j}}}{p^{0}_{j}})^{2}$. ...
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1answer
27 views

Arrange in increasing order of asymptotic complexity

I have to arrange the above time complexity function in increasing order of asymptotic complexity and indicate if there exist functions that belong to the same order. So, my answer is $[lg(n)]^2$ ...
11
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1answer
247 views

Upper bound for the widest matrix with no two subsets of columns with the same vector sum

Over at PPCG there is an ongoing contest going on to find the largest matrix without a certain property, called property $X$. The description is as follows (copied from the question). A circulant ...
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2answers
77 views

Is $x^2+25x+4 \in \mathcal{O}(x^2)$? If yes how? If no why not?

Is $x^2+25x+4 \in \mathcal{O}(x^2)$ ? if yes how ?, if no why? I know $x^2+25x+4\leq 25x^2+25x+25\leq 25x^2+25x^2+25x^2=75x^2$ for some $x$. What confuses me is $x^2+25x+4\leq 25x^3+25x+25\leq ...
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0answers
22 views

Analytic function with inconsistent asymptotic behaviour on rays

Consider an function $f$, defined continuously on the closed upper half plane, and analytic on the upper half plane. Going along any ray from the origin that go strictly up (ie. not along the real ...
5
votes
1answer
107 views

Good resource/exercises for learning asymptotic analysis?

I am studying asymptotic methods right now; things such as mellin transform, inverse mellin transform, saddle point method, laplaces method, etc... and I get very frustrated because I can't get very ...
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1answer
61 views

Why is $(\log n)^3\in O(\sqrt n)$?

Comparing the order of growth of the two functions by taking a limit and using l'hospitals rule, it seems that $\sqrt{n}$ should be O($log^3n$). Here are the steps I took: $$\lim_{n \to ∞} ...
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0answers
19 views

Laplace's Method modifications

I was wondering if there is a "Laplace's Method" to estimate, as $n \to \infty$, integrals of type $$ I_n = \int_0^\infty e^{nh(x)}g(nx) \, dx $$ where $g$ is a smooth function, that converges to a ...
2
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0answers
33 views

Asymptotic of a real double serie on $\mathbb{Z}$

I am interested by a real sequence $\{a_n\}_{n\in\mathbb{Z}}$ as $\sum_{n\in\mathbb{N}}\left(\vert a_n\vert + \vert a_{-n}\vert\right)$ converges. I want to find the asymptotic behavior of this ...
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0answers
9 views

random variables stochastically bound problem

Could you help me about stochastically bound problem for random variable. show that, there exist a sequence of {a_n} of positive real numbers such that X/a_n->0 a.s for any random variable X.
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2answers
61 views

Big theta proof [closed]

a) Show that $3x+7$ is $\Theta(x)$. b) Show that $2x^2 +x -7$ is $\Theta(x^2)$ c) Show that $⎣x+.5⎦$ is $\Theta(x)$ d) Show that $\log_{10}(x)$ is $\Theta(\log_2(x))$ My professor gave the ...
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0answers
10 views

Integral of product of Hermite functions with rescaled weights.

Let $$h_{k}(x)=c_{k}(-1)^k e^{\frac{x^2}{r^2}}\frac{d^k}{dx^k}e^{-\frac{x^2}{r^2}}$$ be the standard Hermite polynomials, rescaled with a given parameter $r>0$. The normalizing constant ...
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0answers
54 views

Verifying an integral identity related to asymptotic homogenization of an elliptic partial differential equation

Background I'm reading Hornung (1997)'s Homogenization and porous media, pg 3: We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, ...
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0answers
19 views

Proof that difference equations as asymptotic to their differential analog.

Given a difference equation $a_{n+k}=f(a_n,a_{n+1},\dots,a_{n+k-1})$, we can classify $n=\infty$ as an ordinary, regular singular, or irregular singular point by classifying $x= \infty$ in the ...
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2answers
73 views

How to show that $\sum_{x=1}^\infty \prod_{i=1}^{x-1} (1-i/n) \sim \sqrt{\frac{\pi n}{2}}$?

How can one show that asymptotically $$\sum_{x=1}^\infty \prod_{i=1}^{x-1} \left(1-\frac{i}{n}\right) \sim \sqrt{\frac{\pi n}{2}} \; ?$$ A non rigorous argument is to say that for large $n$, ...
0
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1answer
35 views

Big O - arithmetic rules

I need to prove the following statement: $O(f(n)g(n))=f(n)O(g(n))$ At first I thought the statement is false but apparently it is true. How can I prove it?
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1answer
70 views

How to show $\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}} \sim 1/n$

How can one compute the large $n$ asymptotics of $$\sum_{i=1}^{n-1} \frac{i(n-2)!}{(n-1-i)!n^{i+1}}\;?$$ My guess is that it is $1/n$ but I don't know how to show that.
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1answer
28 views

Big oh proof for a(n) using big oh hierarcy

So I'm given the following big-oh hierarchy (each sequence is big-oh of any seqeuence to its right.) $1$, $\log_2{n}$, ... , $\sqrt[4]{n}$, $\sqrt[3]{n}$, $\sqrt{n}$, $n\log_2{n}$, $n\sqrt{n}$, ...
4
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1answer
110 views

Growth of ratio based on sum of squared binomial identity

It is a well-known identity that $$\binom{n}{0}^2+\binom{n}{1}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}.$$ By symmetry of the binomial coefficients, this means the ratio ...
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5answers
967 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 infinity slower than all iterations of the (natural) logarithm?
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3answers
143 views

Asymptotic for sum

How can I find formula for $\displaystyle{\sqrt[3]1 + \sqrt[3]2 + \sqrt[3]3 + \cdots + \sqrt[3]n}$ with an accuracy ${\rm O}\left(\, 1 \over \vphantom{\LARGE A}n^{5}\,\right)$ Is here we should use ...
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1answer
21 views

Asymptotic relation between specific binomial coefficient and exponential function

I need to determine the asymptotic relationship between the functions: $$f_1(n)={n\choose{\lfloor{n\over{2}}\rfloor}}, f_2(n)=7^{\sqrt{n}}$$ (I'm going to just assume $n$ is always even.) I've ...
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2answers
48 views

Finding an approximation for the difference of $a_n = \frac{1}{1+a_{n-1}}$ and it's limit.

I've got the recurrence $\displaystyle{a_{n} = {1 \over 1 + a_{n - {\tiny 1}}},\ }$ for $0 < a_{0} < 1 $ which has the solution $\displaystyle{\alpha = {\,\sqrt{\, 5\,}\, - 1 \over 2}}$ I am ...
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0answers
10 views

Asymptotics and function composition

In the following question: Big O and function composition It is explained that if $a, b, c, d$ are functions and $a = O(c), b = O(d)$ it doesn't mean that $a ∘ b = O(c∘d)$. However, what if we allow ...
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2answers
344 views

Inverse of sparse matrix is not generally sparse

I have a question regarding inverse of square sparse matrices(or can be restricted to real symmetric positive definite matrices). I encountered several times the web pages which states that the ...
3
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0answers
38 views

Does $\theta(n)$ = $1/x$ make any sense?

So, I asked this question on a discrete structures exam today, which I apparently didn't give enough thought to: $f(x) = (5x^2 + 6x + 2)/(x^3 + 4x^2 +x)$ Find the correct theta notation for the ...
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0answers
8 views

GRAM series and Logarithmic integral

due to the prime number theorem wouldn't we expect that the prime number counting function admits the approxiamtion $$ \pi (x)= \gamma +loglog(x)+ \sum_{n=1}^{\infty} \frac{log^{n}(x)}{n.n!.\zeta ...
2
votes
1answer
53 views

Number of words not having a subword of length k with only one letter

Let $f_k(n,t)$ be the number of words of length $t$ over the alphabet $\mathcal{A} = \{1,\ldots,n\}$ such that no word contains $i^k$ as a substring for $i \in \mathcal{A}.$ I am looking to find the ...
1
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0answers
33 views

Order of magnitudes comparisons

I need your help with the following. I need to determine how to order (functions) the following : \begin{align} &f(x)=(x/2)^{(x/2)} \\ &g(x)=x! \end{align} Note: I got both of them are ...
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0answers
23 views

Big-O Notation Division

There was a similar thread on this question, but I am still unsure about the answer. I am asked to show, $$ \frac{e^{(r-q)h}-e^{-\sigma\sqrt{h}}}{e^{\sigma\sqrt{h}}-e^{-\sigma\sqrt{h}}} = ...