# Tagged Questions

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

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### Probability of picking each of m elements at least once after n trials.

Suppose I have 10^9 distinct elements, and an equal probability of picking each one in a given trial. How many trials must be conducted for the probability of having picked every element at least once ...
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### How can I prove that Cx will intersect x^2

I want to disprove $cx \geq x^2 \ \forall \ x$ where c is a real number. (i.e. show that x^2 is not O(x) ) So it seems that I can show that the two must intersect at some point ... if I divide both ...
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### Different Upper and Lower Bound

Is there a function or algorithms whose upper bound and lower bound are different? For example f(X) i.e f(X) = O(X^2) and f(X) = Omega(X)
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### Asymptotics of $\sum \sqrt{k}$ and $\sum (-1)^k\sqrt{k}$

I was playing around with series recently and asymptotics of $\sum \sqrt{k}$ and $\sum (-1)^k\sqrt{k}$ were required to solve another problem. I have dealt with the first one using an integral ...
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### Name of the difference between an asymptote and the curve that approaches it

Consider a function, say a hyperbola, and its asymptote. Is there a specific term for the difference between the two? Answers specific to hyperbola, as well as answers about general terminology, are ...
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### Asymptotic Expansion of an Integral involving Modified Bessel Functions

I do not have enough experience with the asymptotic expansion of integrals especially involving Bessel functions. I appreciate any feedback that you guys provide. Here is the problem. Let $a$ and $b$ ...
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### Running Time Question

In what situations would a function of $\theta(n^2)$ perform better than $\theta(n \log n)$? I noticed that in comparing the two, they intersect at $n = 4$. After this, $n \log n$ takes over as ...
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### How to give an upper bound for a solution of $T(n) = T(0.25n) + T(0.75n) + O(n)$?

We have an algorithm which can be described the recurrence formula: $T(n) = T(\frac{n}{4}) + T(\frac{3n}{4}) + O(n)$ and for $n\le 100$: $T(n) = O(1)$. How to show that $T(n) = O(n \log n)$? ...
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### Dominance and Big Oh problem

What is the dominant term in the following expression? 100n + 0.01*(n^2) It is confusing because the power function should be growing faster than the linear function regardless the constants. But ...
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### Algebraic number with bounded coefficients

How many algebraic numbers $z$ are there satisfying $P(z)=0$ where $P(z)$ is some polynomial with integer coefficients of degree less than or equal to $n$ such that the absolute value of every ...
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### Asymptotic solution to inequality $x < k \ln(1+x)$

What is an upper-bound on $x$, given that $x < k \ln(1+x)$? I believe that the solution is something of the form $\mathcal{O}(k \ln k)$ but I am unable to prove this. This is my first encounter ...
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### Find the asymptotes of the Parametric equation?

Consider $$x(t) = 2 e^{-t} + 3e^{2t}$$ $$y(t) = 5 e^{-t} + 2 e^{2t}$$ which represents a non rectilinear paths Horizontal and Verical Asymptotes : If $t \rightarrow +\infty \ \ or \ \ -\infty$, ...
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### Confirm the answer to compute the asymptotic solution to the problem

I have the following problem The solution I derived is $O(g(n))$ where $C = 1, n > 1$. Is this solution correct ?
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### Run time/Efficiency of finding Least Common Multiple

The algorithm is: $$\mathrm{lcm}(x,y)=\frac{xy}{\gcd(x,y)}$$ And we can use the Euclidean algorithm for finding $\gcd$. How is the complexity for above method $O(n^3)$, if $x,y$ can at maximum ...
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### Asymptotic expansion of integral of $e^{-t}/t^n$.

So we study $$f_{n}(x) = \int_x^{+\infty} \! \frac{e^{-t}}{t^{n}} \, \mathrm{d}t, \quad n \in \mathbb{N^{*}}.$$ I've shown that for every $n$, $f_{n}(x) \sim_{+\infty} \frac{e^{-x}}{x^{n}}$. Now I'...
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### Question involving summations and the Θ-notation of running times

I think I understand the concept of summations and Θ-notations, however, I don't really understand the question below. If I have understood it correctly, I'm supposed to write out the summations (...
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### Elementary proof that $\omega(n)$ is bounded $\frac{\log n}{\log( \log n)}$ in the limit?

I'm trying to show that $\omega(n)$ is less than $\frac{\log n}{\log(\log n)}$ as it's stated without proof in an analytic number theory text. It's a corollary of the PNT, but I want to not use that ...
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### Time Complexity for Asymptotic Functions

Here below I have a problem set where I am asked to define the relationship between f(n) and g(n). I have added in my solutions but I wanted to get my answers checked by you guys before I turn this in....
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### Confusion about Big O notation

I have a somewhat stupid question regarding the "Big O" notation: Is there any difference between saying $f=O(g)$ and $f\le O(g)$?
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### recursive-algorithm problem

I am not to sure were to begin Thanks
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### Linking summations with their correct function(s)

Guys can you please guide me step by step on how to link given functions with the functions to choose from. So for example a function $g(n)\in \Theta n^2$ and if there is no match then you say there ...
### Limit and infinite sums. Finding $\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\frac{1}{k^3 x-k^2}$
Could anyone help me with this problem. Compute $$\lim_{x\rightarrow\infty}\sum^{\infty}_{k=1}\dfrac{1}{k^3 x-k^2}$$ I don't know how to change a limit and a sum. Could you help me with this problem ...