# Tagged Questions

18 views

### Is $\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)$?

Is it true that for $n \rightarrow \infty$, $p \gg n^{-1}$, $0<p<1$ and $x=O(1)$, $$\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)?$$
53 views

### Meaning of $\sim$?

I often read $f(x) \sim g(x)$ and I wonder what the Standard Interpretation of this $\sim$ is. It seems to mean something like asymptotically equally distributed, something like $f(x)=g(x)(1+o(1))$. ...
31 views

### Help understanding this approximation

In a paper that I'm reading, the authors write:- $$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04. \tag{4.31}$$ Now, an analytic approximation can be obtained by using the expansion with respect ...
44 views

111 views

### Showing (1 - polynomial fraction) raised to a polynomial power is a negligible function

Let $P(k)$ and $Q(k)$ be two polynomials ($k>0$). Let $\mathrm{neg}(k)$ be a negligible function for sufficiently large $k$ (see Appendix on question for definition). Does someone know how to show ...
34 views

### Check my short proof - asymptotic approximation, which function is bigger

The goal of this exercise is to show that $\ln(n+1)-\ln(n) = O(\frac{1}{n})$ what I did is: I used the fact that if $f=O(g)$ then $\frac{f}{g}=O(1)$. $\ln(n+1)-\ln(n)=\ln(\frac{n+1}{n}) = \ln(O(1))$ ...
40 views

### Asymptotic behaviour of the area of a 2-dimensional flat subset of $\mathbb{R}^3$

I am interested by the area of the $2$-dimensional flat subset of $\mathbb{R}^3$ defined by the following equations with one parameter $t>1$: $x,y,z>0$ (positive octant) $x+y+z=t$ (hyperplane ...
I need to find the asymptotic approximation of this sum $$\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$$ Can you please share a link to theory or hint how it can be solved? Here is my attempt $n ... 1answer 91 views ### Which form of Euler-Maclaurin formula to use? This question may be rather elementary, but I am sort of confused about various forms of the Euler-Maclaurin summation formula and their use. For instance, let us suppose that we want to approximate ... 0answers 44 views ### Asymptotic for Taxicab number. The taxicab numbers are sums of 2 cubes in more than 1 way. First few are - 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, 65728, 110656, 110808, 134379, 149389, 165464, 171288, ... 1answer 328 views ### Bounding the modified Bessel function of the first kind i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ... 3answers 447 views ### the following inequality is true, but I can't prove it The inequality $$\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)$$ holds for all integer$d\geq 1$. I use computer to verify it for$d\leq 50$, and find ... 1answer 80 views ### Weighted sum of ratio of partial sum of binomial coefficients I would like to approximate the following sum when$n \rightarrow \infty$and$n \gg k\$, $$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose ... 2answers 63 views ### Roots of the equation I_1(b x) - x I_0(b x) = 0 I'm interested in the roots of the equation: I_1(bx) - x I_0(bx) = 0 Where I_n(x) is the modified Bessel function of the first kind and b is real positive constant. More specifically, I'm ... 3answers 227 views ### “O” notation in Stirling approximation In the Stirling approximation the formula as typically used in applications is$$\ln n! = n\ln n - n +O(\ln(n))$$I'm confused about the last term "O" . What does it mean actually, and when do we ... 1answer 584 views ### Method of dominant balance Find the leading asymptotic behaviour as x \rightarrow \infty of$$x^2y'' + (1 + 3x)y' + y = 0 $$Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ... 1answer 208 views ### Techniques for bounding a sum I have a very messy function. It consists sums four levels deep, and the inner-most term is itself quite messy.$$ \sum \sum \sum \sum (\mbox{stuff})$$I can't find a closed form for this function. ... 0answers 114 views ### When does distribution bootstrap mean converge to distribution sample mean? Let \bar{X}_n denote the sample mean of n iid random variables. Let \bar{X}^*_n be the bootstrap sample mean. Does \left|\mathbb{P}\left(n^{1/2}\bar{X}_n\leq ... 1answer 187 views ### Laplace's method with unknown exponent. Given the integral:$$I = \int_0^a{e^{-\lambda g(x)}f(x)dx}$$Where g(x) and f(x) are both low order positive polynomials, and \lambda \gg 1, Laplace's method is commonly used to approximate ... 1answer 78 views ### Finding the asymptotic limit of an integral. I'm having trouble finding the asymptotic of the integral$$ \int^{1}_{0} \ln^\lambda \frac{1}{x} dx$$as \lambda \rightarrow + \infty. Can anyone help? Thank you! 0answers 76 views ### Describe growth of \epsilon n For all \epsilon we have that f(n)\le \epsilon n where n is a natural number. What can we say about the growth of f(n)? Clearly f(n)=O(n), can we say anything sharper? 1answer 110 views ### asymptotic limit of \int_0^{\infty}\left(1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}\right)^qdt Help me please with the following integral. I've asked this question before Asymptotic limit of the integral with polynomial, but it turns out that it was incorrect question. I should get an ... 1answer 75 views ### asymptotic limit at the integral I would like to get an asymptotic limit at the following integral: for p\ge 2, n \in N, t \ge 0$$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$I think ... 0answers 65 views ### On bounding the average cost of top-down merge sort Let A_n be the average number of comparisons to sort n keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that$$ A_0 = A_1 = 0;\quad A_n = ...
Stirling approximation to a factorial is $$n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.$$ I wonder what benefit can be got from it? From computational perspective (I admit I don't ...