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What is the O-notation (or $\Theta$ notation ) of $\left(\begin{array}{c} n\\ \frac{n}{2} \end{array} \right)$ ?
Can I use Sterling approximation : $n! = \Theta(\sqrt{n}\left(\frac{n}{e}\right)^n)$ and evaluate my function as $\Theta\left(\frac{2^n}{\sqrt{n}}\right)$ ?

Eliran Turgeman
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    Why wouldn't you? Just be careful to do the "arithmetics" correctly. I'd recommend to 1) apply the definition of the binomial, then use the $\sim$-asymptotic for factorials, and 3) use $\lim f/g$ with some lemmas to prove your claim. Our reference question has some material. – Raphael Apr 05 '20 at 14:44
  • I'm just a bit unsure of how much carefulesness I need to apply.
    will this calculation suffice?
    $$\lim_{n \to \infty} \frac{\frac{n}{2}!}{ {\sqrt{\frac{n}{2}}\left(\frac{n}{2e}\right)^{n/2}}} \in (0, \infty)$$     – Eliran Turgeman Apr 05 '20 at 15:17
  • That's not a calculation. If you can edit the question to add your whole attempt, people can have a look. – Raphael Apr 05 '20 at 17:26
  • Regarding carefulness, something that doesn't work is to set up $\lim f/g$, then replace $f$ and $g$ with $\Theta$-asymptotics and start doing arithmetics; you're working towards a constant, but Landau notation hides those. Using $\sim$-asymptotics, on the other hand, just works (maybe citing a suitable lemma). – Raphael Apr 05 '20 at 17:29
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    We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. – D.W. Apr 05 '20 at 17:36

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