# Tagged Questions

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

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### On corollary and theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
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### On “bounded” in intuition for a theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
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### Big-O proof of inclusion

I'm working on this proof of inclusion:$$\log_2(8^n)\in{\mathcal O(n)}$$ $$\log_28^n-cn\leq0$$ for all $n>n_0$. Is there a log rule that I can use to further simplify before I plug random values to ...
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### How to calculate $O(\sum_{k=1}^{K}(N-k)(k+1)^2)$?

Using the formula for the sum of the squares and the sum of first $K$ numbers I can get that: $$\sum_{k=1}^{K}(N-k)(k+1)^2=\dfrac{1}{12}K(-3K^2+2K^2(2N-7)+3K(6N-7)+26N-10)$$ Now I guess I can ...
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### asymptotic expansion of an expression involving modified bessel function

I am looking for the asymptotic behavior of $$g(t,\nu)=e^{-t^2}\left[I_\nu(t^2)+\frac{1}{2}\left(I_{\nu+1}(t^2)+I_{\nu-1}(t^2)\right)\right]$$ as $t\rightarrow \infty$. Here $\nu$ only takes even ...
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### Proportion of elements of prime order $p$ in $S_n$

I was trying to answer the following question recently : What is the proportion of elements of order $p$ in the symmetric group $S_n$ , where $p$ is some prime number ? I managed to work out that in ...
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### Does any quadratic function in the form $an^2 + bn + c$ equal $\Theta(n^2)$ in asymptotic notation?

On a Khan Academy post (see here) about Big-$\Theta$ notation, the author attempted to convert the quadratic function $6n^2 + 100n + 300$ to asymptotic notation. They started by dropping the $n^2$ ...
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### Sum of first n primes [duplicate]

Can we claim it is asymptotic to $n^2\log n$? I argue that because $p_n\sim n\log n$, we can say: $$\sum_n n\log n=\log1+2\log2+\dots+n\log n$$ $$=\log1+\log2+\dots+\log n$$ $$+\log2+\dots+\log n$$ ...
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