# Tagged Questions

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

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### Why is $\lim\limits_{x \to +\infty}\frac{x \sqrt{x+2}}{\sqrt{x+1}} - x = \frac12$?

I need to evaluate this limit: $$\lim_{x \to +\infty}\frac{x \sqrt{x+2}}{\sqrt{x+1}}-x$$ to calculate the asymptote of this function: $$\frac{x \sqrt{x+2}}{\sqrt{x+1}}$$ which, according to the class ...
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### Hypergeometric function how to detect blow up for large arguments

I have to study 11 solutions to ODEs containing multiple hypergeometric functions of type: $${}_2F_2 \left(\begin{array}{c} a_1,a_2 \\ b_1,b_2 \\ \end{array} ;z \right).$$ My main ...
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### Show that $(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\tfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\tfrac{1}{n^{3/2}} \right)$

I would like to show that : $$\fbox{(-1)^{n}\left(\sqrt{n+1}-\sqrt{n} \right)=\dfrac{(-1)^{n}}{2\sqrt{n}}+\mathcal{O}\left(\dfrac{1}{n^{3/2}} \right)}$$ by starting from the left side and get ...
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### Can two function be Big-O of each other?

Given two functions $f(n)$ and $g(n)$, is it possible that $f(n) = O(g)$ and that $g(n) = O(f)$? If the answer is yes, I have a follow-up: if $f(n)$ and $g(n)$ are Big-O of each other, does that ...
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On the wiki page for forward euler (https://en.wikipedia.org/wiki/Euler_method#Local_truncation_error), it describes the local truncation error like so: $\mathrm{LTE} = y(t_0 + h) - y_1 = \frac{1}{2} ... 1answer 27 views ### Can you provide us an asymptotic for this series involving Mertens functions? Let for integers$k\geq 1$, the Möbius function denoted by$\mu(k)$, and$M(n)=\sum_{k\leq n}\mu(k)$the Mertens function, then one can prove easily that $$\sum_{k=1}^n\mu(k)\frac{e^{\mu(k)}+1}{e^{\... 2answers 55 views ### Asymptotics of f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} Define$$ f_c(n)=\sum_{k=0}^{\lfloor cn\rfloor}{n\choose k} $$for some fixed constant c (say, 0<c<1/2). What are the asymptotics of f_c(n) as n\to\infty? It seems that this should be ... 1answer 89 views ### Counting function for sums of three squares Legendre showed that an integer is the sum of three squares if and only if it is not of the form 4^n(8m + 7) for some nonnegative integers n and m. However, I have been unable to find any ... 0answers 33 views ### Asymptotic expansion of elliptic integral I am trying to find the first 2-3 terms of the asymptotic expansion in terms of 1/ρ of the elliptic integral I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\... 3answers 128 views ### What is the value of I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.? Find the integral I.....it looks like a good problem which I was not able to solve ....please help...$$I=\lim_{n \to \infty} \int_0^1 {{1 + nx^2}\over{(1 + x^2)^n}} \log(2 + \cos(x/n))\,dx.$$3answers 1k views ### Running time (Big O) of counting in binary What is the total running time of counting from 1 to n in binary if the time needed to add 1 to the current number i is proportional to the number of bits in the binary expansion of i that must ... 0answers 23 views ### When is \frac{2 n f(n)}{n !} in the order of some fixed power of n? I would like to know when \frac{2 n f(n)}{n !} is O (n^b) where b is a constant. Here, n is a positive integer. My attempt:$$ \frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{... 1answer 59 views ### Show :$(-1)^{n}n^{-\tan\left(\tfrac{\pi}{4}+\tfrac{1}{n} \right)}=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)$I would like to show that : $$(-1)^{n}n^{-\tan\left(\dfrac{\pi}{4}+\dfrac{1}{n} \right)}=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ My proof: Note that : \begin{... 0answers 19 views ### Show that$\tfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\tfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \tfrac{1}{n^{\tfrac{3}{2}}}\right) $I would like to show that : $$\dfrac{(-1)^{n}}{\cos(n)+n^{\tfrac{3}{4}}}=\dfrac{(-1)^{n}}{n^{\tfrac{3}{4}}}+\mathcal{O}\left( \dfrac{1}{n^{\tfrac{3}{2}}}\right)$$ by starting from the left side ... 0answers 48 views ### Show that$(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\tfrac{(-1)^{n}}{n}+\mathcal{O}\left(\tfrac{\ln(n)}{n^{2}} \right)\$

I would like to show that : $$(-1)^{n}\sqrt[n]{n}\sin(\frac{1}{n})=\dfrac{(-1)^{n}}{n}+\mathcal{O}\left(\dfrac{\ln(n)}{n^{2}} \right)$$ by starting from the left side and get the right side : My ...