# Tagged Questions

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

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### Asymptotic behaviour of sum over the inverse japanese symbol

I am interested in the asymptotic behavior of the sum $$\sum_{m=1}^M\frac{1}{\sqrt{m^2+\omega}}$$ for $1>\omega>0$ in the Limit $M\to\infty$ up to order $\mathcal{O}(M^{-1})$. The first thing I ...
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### Difference between the definitions regarding distribution of prime numbers

Following are the two theorems that Hardy and Wright state in their book Theorem A: The number of primes not exceeding $x$ is given by $\pi(x) \sim \frac{x}{\log{x}}$. Theorem B: The order ...
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### Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me. I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes. One is the ...
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### Finding a degree-2 polynomial that sits under the Harmonic Numbers.

Does there exist a degree-2 polynomial with positive acceleration such that the real extension of the harmonic numbers surpasses it for all future values? This was too big of a title and it's a ...
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### How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$

Determinate the best asymptotic expansion for $x \to 0$ for: $$\sqrt{1-2x+x^2+o(x^3)}$$ How should I procede? In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to ...
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### How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
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### Mittag Leffler Stability [closed]

How to prove "Mittag-Leffler stability implies asymptotical stability" ? What should be done? Thanks.
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### What is $\int_{\Omega'} \psi (\nabla p) dV \: \text{as} \: \delta\alpha \rightarrow 0$?

I have an axi-symmetric integral (the domain and all functions are axi-symmetric) in cylindrical coordinates which needs to be integrated by parts for use in a finite element code. The integral is ...
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### Asymptotics of incomplete Beta function $B_{1/2}(y+1,y)$ when $y\to\infty$

My question concerns the behavior of the incomplete Beta function $$B_{1/2}(y+1,y)=\int_0^{1/2}x^y (1-x)^{y-1}dx$$ in the large $y$ limit. I have been looking everywhere, but I can't find anything. ...
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### Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
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### Testing convergence of series $\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$ [duplicate]

Considering $$\sum_{n=1}^\infty\sin(\pi\sqrt{n^2+k^2})$$ depending on $k$, which can be real. I have absolutely no clue how to proceed. Tried to taylor it, but with no result.
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### Product of Fibonacci numbers

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers. Does there exist a tight approximation for these kind of things?
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### How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ...
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### When is $1-(1-p)^n \sim pn$

Let $0<p=p(n)<1$ with $p=o(1)$. For which $p$ is it true that $1-(1-p)^n \sim pn$? With $\sim$ I mean that they are asymptotically the same, so $\frac{1-(1-p)^n}{pn}\rightarrow 1$, or at least ...
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### What is the purpose of Stirling's approximation to a factorial?

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 ...
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### How quickly can we find a value that has large multiplicative order modulo $n$?

If we're trying to find an element modulo $n$ that has multiplicative order at least $\sqrt{n}$, how quickly can we do this? We don't know if $n$ is prime or composite, only that $n$ definitely has a ...
### $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$
Consider $(x + y + xy)/2 = f( f^{[-1]}(x) + f^{[-1]}(y) )$ Where $f^{[-1}]$ denotes the functional inverse of $f$. How to find $f$ ? How about the more General idea of finding $f$ for a given $g$? ...