Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [asymptotics]

For questions involving asymptotic analysis, including function growth, Big-$O$, Big-$\Omega$ and Big-$\Theta$ notations.

Questions involving asymptotic analysis, including function growth, Big-$O$, Big-$\Omega$ and Big-$\Theta$ notations.

  • $f(x) = O(g(x))$ as $x \to \infty$ is used to mean that for sufficiently large values of $x$, we have $|f(x)| \leq A g(x)$ for some constant $A$.

  • $f(x)=\Omega(g(x))$ is equivalent to saying that $g(x)=O(f(x))$.

  • $f(x)=\Theta(g(x))$ is used to mean that $f(x)=O(g(x))$ and that $f(x)=\Omega(g(x))$.

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Prove that $\log(n) = O(\sqrt{n})$

How to prove $\log(n) = O(\sqrt{n})$? How do I find the $c$ and the $n_0$? I understand to start, I need to find something that $\log(n)$ is smaller to, but I m having a hard time coming up with the example.
Anson
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How to show that $\sum\limits_{k=1}^{n-1}\frac{k!k^{n-k}}{n!}$ is asymptotically $\sqrt{\frac{\pi n}{2}}$?

According to "Concrete Mathematics" on page 434, elementary asymptotic methods show that $\displaystyle \sum_{k=1}^{n-1}\frac{k! \; k^{n-k}}{n!}$ is asymptotically $\sqrt{\frac{\pi n}{2}}$. Does anybody see how to show this?
PeterR
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Asymptotics of $1^n + 2^{n-1} + 3^{n-2} +\cdots + (n-1)^2 + n^1$

Suppose $n\in\mathbb{Z}$ and $n > 0$. Let $$H_n = 1^n + 2^{n-1} + 3^{n-2} +\cdots + (n-1)^2 + n^1.$$ I would like to find a Big O bound for $H_n$. A Big $\Theta$ result would be even better.
qwerty89
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How do you prove that $n^n$ is $O(n!^2)$?

It seems obvious that: $$n^n \in O(n!^2)$$ But I can't seem to find a good way to prove it.
Guillaume Martres
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What's the lower bound of the sum $S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n)$?

If we have $$ S(n) = \sum_{k=1}^n \prod_{j=1}^k(1-\frac j n) $$ What the lower bound of $S(n)$ when $n\to\infty$? PS: If I didn't make any mistake when I calculate $S(n)$, then it should be $\Omega(n)$. But I don't know how to get it.
NonalcoholicBeer
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15
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A recurrence that wiggles?

Consider the following sequence $a_n$: $a_1 = 0$ $a_n = 1 + \frac{1}{2^n-2} \sum_{i=1}^{n-1} \binom{n}{i} a_i$ The first few terms are $0,1,\frac{3}{2},\frac{13}{7},\frac{15}{7}$. The sequence comes out of the analysis of a certain process, whose…
Yuval Filmus
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Asymptotics of Sums

Suppose $f(x)$ is a function such that $f(x)=O(x)$, then $$\sum_{n\leq x} f\left(\frac{x}{n}\right) =O\left( \sum_{n\leq x} \frac{x}{n}\right) = O\left(x\sum_{n\leq x}\frac{1}{n} \right) = O\left( x\log(x)\right)$$ Conversely, if $g(x)$ is an…
Mastrel
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Sum of Big O - which one is it?

On the wikipedia page, we have the following property: If $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ then $f_1(x) + f_2(x) = O(|g_1(x)| + |g_2(x)|)$. But in my textbook, I also see the following sum property: If $f_1(x) = O(g_1(x))$ and $f_2(x)…
user56031
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How to Use Big O Notation

In my question about the convergence/divergence of $$ \sum_{n=2}^\infty \frac{1\cdot 3\cdot 5\cdot 7\cdots (2n-3)}{2^nn!}. $$ here: Why Doesn't This Series Converge? Zarrax gave the answer: "You can use Taylor approximations here. Note that the…
J126
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Estimating the integral $\int_0^1 (1-t^2)^{-1/2} e^{-nt} \,dt$ for large $n$.

I would like to find the asymptotic behavior of the integral $$\int_0^1 (1-t^2)^{-1/2} e^{-nt} \,dt$$ for large $n$. It seems reasonably obvious that the integral goes to zero. At least it is bounded; the integral is between $0$ and $$\int_0^1…
Antonio Vargas
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Functions between polynomial and exponential

Does there exist a function $f(n)$ such that as $n \rightarrow \infty$, we have $p(n) < f(n) < e(n)$? Where $p$ is any polynomial and $e$ is any exponential (e.g. $e(n) = e^{\alpha n}, \alpha > 0$)
countunique
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10
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Is there a function such that $f(f(n)) = 2^n$?

In this question, I was looking for a specific "middle family" of functions between polynomials and "anti-polynomial exponentials", as I will call them, which are functions like like $2^{\sqrt{n}}$ that become exponential when chained to a…
Joe Z.
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Prove that $3^n$ is not $O(2^n)$

I have this question in my assignment. I need to prove, using only the definition of $O(\cdot)$, that $3^n$ is not $O(2^n)$. It is obviously true for any $n \geq 1$. To prove $3^n \in O(2^n)$, we must find $n_0$, $c$ such that $f(n) \leq c \cdot…
Justin D.
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Estimate $\sum_{k=1}^{n} k^{k-1} \binom{n}{k} (n-k)^{n+1-k}$

I'm interested in estimating $$X_n=\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n+1-k}$$ up to and including terms of order $n^n$; that is, I want $f_n$ in $X_n=f_n+o\left(n^n\right)$. The following identity looks very similar but I am not sure…
user66151
9
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1 answer

Complexity $O(n^3)$ vs $O((\log n)^4))$

I would like to prove that $O(n^3)$ is bigger than $O((\log n)^4)$. I thought that I can divide both powers with 4 so it is $$O\left(n^{\frac{3}{4}}\right)$$ vs $$O(\log n)$$ but then I don't know how I can prove that $$O(n^k)$$ is bigger than…
Complex1238877
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