# Tagged Questions

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

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### Showing $n^{\log{n}} = o(2^n)$

I would like to show that $n^{log n} = o(2^n)$. Here is my attempt: I see that $\log{(n^{\log{n}})} = (\log{n})^2,$ and $\log{2^n} = n\log{2}$. I also know that $(\log{n})^2=o(n)$, so that for ...
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### Orders of Asymptotes

We know that $\log(X)^n = o(X^\epsilon)$ for all $n,\epsilon>0$. My questions is, is $\log(X)$ the largest function that is smaller than all (small) powers of $X$. That is, can we find a (non-...
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### On finding the order of an infinitely small quantity

Given an infinitely small quantity: $$\alpha \left ( x \right )= \tan \left ( x \right )-\sin \left( x \right)$$ as x aproaches $0$, and computing the corresponding asymptotic relationship. What does ...
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### Can we give a bound on any associative function?

We say that $f:[1,\infty)^2\to[1,\infty)$ is associative if $$f(f(a,b),c)=f(a,f(b,c))$$ And symmetric if $$f(a,b)=f(b,a)$$ e.g. the arithmetic operations '+' and '$\cdot$' are associative and ...
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### How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ we have $$\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} \,dt = O(z^{-1} )?$$ According to Serge Lang, the integral on the left is the error term for ...
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### Asymptotic form of the integral $\int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs}$ for $s \to \infty$

I would like to find an asymptotic form of the following integral when $s \to \infty$ ($s$ and $w$ are positive) $$\int_{0}^{\infty} dx ~ \sqrt{x^2 + wx} ~ e^{-ixs}$$ I ...
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### Closed form for $\prod_{k=1}^n (a+k^2)$

I have come across the following product: $$\prod_{k=1}^n (a+k^2)$$ where $a$ is a positive constant. Could anyone suggest a closed form for this product? I need to approximate this for large $n$, ...
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### Asymptotic expansion of the complete elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k) = \int_0^{\pi/2} \frac{d x}{\sqrt{1 - k^2 \sin^2 x}}.$$ I would like to derive (at least the first term of) the asymptotic ...
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### $f(x)\sim 1/x \implies (1+f(x))^x\to e$, but what family of functions maximizes the speed of convergence from below?

This problem is subordinate to finding out if $$\left(1+\frac{\log p_{n+1}}{p_n}\right)^{p_{n+1}/\log p_n},$$where $p_n$ is the $n$-th prime, never stabilizes above or below its limiting value, which ...
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### How to bound the tail of p-series

How can I asses $S_n = \sum_{j=n}^\infty\frac{1}{j^p}, p>1$ in terms of $n$, specifically can I get something like $$S_n = O(?)$$
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### How to prove $\omega$ bound without using limit?

How to show $n^{3.4} - 2015n^{2} + 3$ $\in$ $\omega(n^{3})$ without using limit? According to the definition of $\omega$, $f(n)$ $\in$ $\omega(g(n))$ if and only if $\forall c > 0$, $\exists n_0$ ...
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### $u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

how can i show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
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### Asymptotic Growth: little o(n) versus $O(n^\alpha)$

Let $f(n) \geq 0$ be defined for all $n \in \mathbb{N}$. Suppose $f(n)$ is $o(n)$ and at the same time $f(n)$ is not $O(n^\alpha)$ for all $0 \leq \alpha < 1$. Is this necessarily a contradiction? ...