# Questions tagged [ceiling-function]

The ceiling function maps a real number $x$ to the smallest integer greater than or equal to $x$, often denoted $\lceil x\rceil$. See also (floor-function).

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### How to find the partial sum when a series uses $\operatorname{ceil}$?

Take this partial sum. By the formula for a finite geometric series: $$\sum_{k=1}^n2\pi^{k-1}=2\left(\frac{1-\pi^n}{1-r}\right)$$ Simple enough. However, when a $\operatorname{ceil}$ is introduced,...
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### What is the asymptotic upper bound of a variable in this functional equation?

We are given a recursive function $f(x) = \lceil \frac {f(x + 1)}{\lceil \log_2(f(x + 1)) \rceil} \rceil$. We know that $f(1) = 2$ and $f(a) = n$. What is the asymptotic upper bound of $a$ expressed ...
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### $\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$ when $a,b$ are integers? [closed]

Let $a$ and $b$ be positive integers. If $b$ is even, then we have $$\left\lfloor \frac{a-b}{2} \right\rfloor + \left\lceil \frac{a+b}{2} \right\rceil = a$$ I think the equality also hold when $b$ ...
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### Equivalence between ceil and floor functions

I was reading heap data structures from various sources. They used to explain heap as stored in array. One source has array starting at index 0. Other has it starting at 1. They specify different ...
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### Is it possible to represent this function as a polynomial, by removing the ceiling function?

I've been working through a derivation and have arrived at the following expression: $$E = 1 - \frac{x}y \left( \bigg\lceil \dfrac{x}{y} \bigg\rceil \right)^{-1}$$ where $x,y \in \mathbb{R^+}$. I ...
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### Are there infintely many primes generated by the recursion $c_{n+1} = \lceil \frac{3}{2} c_{n} \rceil$?

Inspired by a recent discussion (How to solve a ceiling expression or recurrence equation?) I stumbled on the question: Are there infinitely many primes in power ceiling series? If not there must ...
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### is this statement about x, y, and z true?

I would like to know if the following statement about x,y and z is true: $$x=\lfloor\frac{y}{z}\rfloor \iff z=\lfloor\frac{y}{x}\rfloor$$ I think it is true but am having a hard time wrapping my ...
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### Calculate $\int_0^1{x·\lceil1/x\rceil dx}$

I am trying to calculate following integral: $$\int_0^1{x·\biggl\lceil \frac{1}{x}\biggr\rceil dx}$$ I tried usual change t=1/x but not able to further advance. Thanks!
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### Prove that $\lfloor x\rfloor \geq y$ if, and only if, $x\geq\lceil y\rceil$

I have some trouble proving that if $x,y\in\mathbb{R}$ then $\lfloor x\rfloor \geq y$ if, and only if, $x\geq\lceil y\rceil$. I have tried some different approaches, the most recent being a proof by ...
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### Optimization with ceiling function to determine voxel size

I am trying to calculate the voxel size for a voxel grid which must enclose a $3$D object with real dimensions $\alpha$,$\beta$,$\gamma$ ($\ge 0$). The amount of voxels may be at most $\theta$ (...
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### How can I express this expression without ceiling function: $\left\lceil \frac {1003}{3000}×2^{2n-1} \right\rceil$

I think the following equality is correct for $n\in \mathbb{Z^{+}}$ $$\left\lceil \frac 13×2^{2n-1} \right\rceil=\frac 13×(2^{2n-1}+1)$$ Now, I need to find such a equality for the following ...
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