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 $f(x)=\Omega(g(x))$.

9469 questions
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Solve $\sqrt{1-2x+x^2+o(x^3)}$ with $x \to 0$

I need help to solve $\sqrt{1-2x+x^2+o(x^3)}$ with $x \to 0$, I do not understand when and why I should stop. Here my steps: I can use Taylor formula for $\sqrt{1+t}$ so: $$\sqrt{1+t} = 1+ \frac{t}{2}-\frac{t^2}{8}+\frac{t^3}{16}+o(t^3)$$ where: $t…
Christian Giupponi
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Why is this big oh $O(n^3)$

Why is this big oh $O(n^3)$? (b) Give a good big-Oh bound on the function $$f(n)=2^{\log_2 n} n^2 + 3n^2 \log_2 n +n -17$$ I am not sure on how to solve this. If someone could help me solve, I will be thankful as I have an exam tomorrow.
Maddy
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Asymptotic analysis : Theory

how do you prove that when the limit of n approaches towards positive infinity while n^2/(log n)! We tried to used Stirling theorem but this may not work due to the fact that it may or may not exist on all intervals.
Darko Atanackovic
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2 answers

Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$

Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$ I think I would start with expanding the left side. How would I go about this?
User
  • 405
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1 answer

Asymptotic behaviour of $\ln( \binom{n^3}{2n^2} )$

Given $\ln( \binom{n^3}{2n^2} )$. Determining asymptotic behaviour I have got next result: $$ \ln( \binom{n^3}{2n^2} ) = \ln( \frac{(n^3)!}{(2n^2)! (n^3 - 2n^2)! } ) \approx $$ $$ \approx 2n^2\ln{n} + O(n^2) $$ Is my result right for asymptotic…
J.Exactor
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2 answers

Arrange the following growth rates in increasing order O((35)^n),O(n^4),O(1),O(n^3 logn)

I want to Arrange the following growth rates in increasing order This order are following : O((35)^n),O(n^4),O(1),O(n^3 logn) Please give me idea how to arrange growth rates in increasing order
Rahul Vijayaragavan
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2 answers

Prove $\frac{1}{(1+z)^2}=1-2z + \mathcal{O}(z^2)$ as $z \to 0$

Prove: $$\frac{1}{(1+z)^2}=1-2z+\mathcal{O}(z^2)$$ as $z\to 0$.
Aleksandar
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2 answers

How to prove order of equation using Big-Oh notation?

How can I prove this order equation using Big-Oh notation? $$O(3n^3+2n^2+5) = n^3$$
talha06
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Big Omega equation

I am struggling still with this equations...from my class materials.... This time we deal with lower bound -> BIG OMEGA: I know that: $$\Omega(g(n)) = \{f(n) : \exists c, n_0 > 0\,\forall n\ge n_0\ \ cg(n) \le f(n)\}$$ 1) $\frac 13 n^2 − 3n =…
forrestGump
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3 answers

Prove $(\log_2n)^{100} = \mathcal O(n^{1/10})$

On my homework: prove that $$(\log_2n)^{100} = \mathcal O(n^{1/10})$$ Any ideas are appreciated.
Allan Jiang
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1 answer

Simple question about big O

If $f(n)=g(n)$, can we just say that $\mathcal{O}(f(n))=\mathcal{O}(g(n))$? ($f$ and $g$ are two $\log$ functions) Is it definitely yes? if not please describe why.
user3576199
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Find a two-term asymptotic expansion of $xe^{-x} = \epsilon$.

I want to find a two-term asymptotic expansion of the equation above. For the small root, I know the first term may be $\epsilon$, but I don't know how to find the second term. Moreover, I don't know how to find the root tending to infinite either.
ZYZmoonlight
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1 answer

Equivalent of $\tan(\frac{\pi}{2x+1})$ in zero

My task is to find the equivalent of $$\tan\left (\frac{\pi}{2x+1}\right )$$ in zero. I tried using the formula $\tan(x)\sim x$ in zero, and got $\frac{\pi}{2x+1}$ and then this is $\sim\pi$ in zero. But according to the teacher the right answer is…
Loukil-M
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2 answers

What if $a_n = o(1)$ and $a_n = O(1)$?

What happens when $a_n = o(1)$ and $a_n = O(1)$? What can we say about $a_n$ and its growth rate?
Sofija
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Big O and equality

On this problem, I'm not sure what Big O definition they are referring. How would the big o definition help show this? I get that $|y-y_h|$< $M\beta (h)$, where $y_h$ is an approximate value to y at some point h, but don't really understand the…
john
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