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Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

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Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $

I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one: DEFINITION Given $x\in \Bbb R$, the integer ...
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How to solve an definite integral of floor valute function?

How do you prove this identity: $$\int_0^{n^2}\lfloor\sqrt{t}\rfloor dt = \frac{1}{6}n(n-1)(4n+1)$$ I'd very much appreciate your help on this one!
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Proof of greatest integer theorem: floor function is well-defined

I have to prove that $$\forall x \in \mathbb{R},\exists\text{ exactly ONE }n \in \mathbb{Z} \text{ s.t. }n \leq x < n+1\;.$$ I'm done with proving that there are at least one integers for the ...
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How to find $\sum_{i=1}^n\left\lfloor i\sqrt{2}\right\rfloor$ A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).

[A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).][1] If $n = 5$ then $$\left\lfloor1\sqrt{2}\right\rfloor+ \left\lfloor2\sqrt{2}\right\rfloor + \left\lfloor3\sqrt{2}\right\rfloor +\left\lfloor4 \...
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How do we prove that $\lfloor0.999\cdots\rfloor = \lfloor 1 \rfloor$?

Are the floor functions of $0.999\cdots$ and 1 equal? It is true that $0.999\cdots=1$ but how does one justifies the integer part of $0.999\cdots$ being 1 , where it is not, or alternatively without ...
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Solve summation $\sum_{i=1}^n \lfloor e\cdot i \rfloor $

How to solve $$\sum_{i=1}^n \lfloor e\cdot i \rfloor $$ For a given $n$. For example, if $n=3$, then the answer is $15$, and it's doable by hand. But for larger $n$ (Such as $10^{1000}$) it gets ...
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For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried ...
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How do I evaluate this sum(involving the floor function)? [duplicate]

$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
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Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$. Challenge: Prove $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\...
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Find a formula for $\sum\limits_{k=1}^n \lfloor \sqrt{k} \rfloor$

I need to find a clear formula (without summation) for the following sum: $$\sum\limits_{k=1}^n \lfloor \sqrt{k} \rfloor$$ Well, the first few elements look like this: $1,1,1,2,2,2,2,2,3,3,3,...$ ...
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How to find $\lfloor 1/\sqrt{1}+1/\sqrt{2}+\dots+1/\sqrt{100}\rfloor $ without a calculator?

$$ \left\lfloor\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} +\dots+ \frac{1}{\sqrt{100}}\right\rfloor =\,? $$ I rationalized the denominator and then I think I should somehow group the numbers, but i don'...
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Proving that $x$ is irrational if $x-\lfloor x \rfloor + \frac1x - \left\lfloor \frac1x \right\rfloor = 1$

Prove : $$ \text{If } \; x-\lfloor x \rfloor + \frac{1}{x} - \left\lfloor \frac{1}{x} \right\rfloor = 1 \text{, then } x \text{ is irrational.}$$ I think the way to go here is to falsely assume that $...
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Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$ [duplicate]

Right, this is an exercise in Apostol, which I am not being able to solve. I was able to prove this result for a small case, that is the case when $n=2$, $[x] + \Bigl[x + \frac{1}{2}\Bigr]=[2x]$, but ...
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A valid floor function trick?

Given $x\in\mathbb R_+$ and $m,n\in\mathbb Z_+$, is it true that $$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$$ Thanks for at least three ...
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Proving $\lfloor 2x \rfloor = \lfloor x \rfloor + \lfloor x+0.5\rfloor$

I can intuitively see that this is true, but I'm having a very hard time proving it. I'm actually not even quite sure where to begin. I tried using the inequalities that define the floor function, and ...
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Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$

Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$ I'm trying to solve it by using the squeeze theorem but I'm stuck. I'm looking for a function $g(x)$ such that $g(x) \leq x\lfloor\frac{1}{x}\...
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Verifying a Superior Integration Method for Step Functions

I should note that this is an integration algorithm and therefore intermediate steps DO appear to be unjustified. The purpose of this question is to justify or reject this algorithm as always giving ...
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How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \lfloor\frac{\lfloor x\rfloor}{n}\rfloor=\lfloor\frac{x}{n}\rfloor$.

How to prove or disprove $\forall x\in\Bbb{R}, \forall n\in\Bbb{N},n\gt 0\implies \left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor=\left\lfloor\frac{x}{n}\right\rfloor$. So we want to prove $\left\...
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Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$

Via induction I need to prove an expression is true. the expression is: $n \leq 2^k \longrightarrow a_n \leq 3 \cdot k2^k + 4 \cdot 2^k-1$ for all $n,k \in \mathbb{Z^+}$ $a_n$ is a recursive ...
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1answer
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Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$

So this question in my book looks like it's essentially asking me to prove the ceiling function exists. This question is slightly different to other things I found in related question because we're ...
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Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?

Our algebra teacher usually gives us a paper of $20-30$ questions for our homework. But each week, he tells us to do all the questions which their number is on a specific form. For example, last ...
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there exist infinite many $n\in\mathbb{N}$ such that $S_n-[S_n]<\frac{1}{n^2}$

recent conjecture :Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that :there exist infinite many $n\in\mathbb{N^{+}}$ such that $$S_n-[S_n]<\dfrac{1}...
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Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed $$\sum_{k=...
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Does $\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right)=a?$

Does $\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \rfloor\right)=a?$ I'm going to say this statement is false, and try to use the properties of limits $$\lim_{x \to 0+} \left(x\lfloor \frac{a}{x} \...
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Sum of $\lfloor k^{1/3} \rfloor$

I am faced with the following sum: $$\sum_{k=0}^m \lfloor k^{1/3} \rfloor$$ Where $m$ is a positive integer. I have determined a formula for the last couple of terms such that $\lfloor n^{1/3} \...
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For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

This is Exercise 3.20 from Apostol's book. Many of them seem tough and here is one of them which I am struggling with. For $n \in \mathbb{N}$, prove that this identity is true: $$\Bigl\lfloor{\sqrt{...
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Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$. [closed]

Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$. How should I solve this? I can't think of a way with ...
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How to find $\lim_{x \to \infty} [x]/x$?

Find the limit $$\lim_{x\to \infty } \ \frac {[x]}{x}.$$ Does $[x]$ means greatest integer in this case?
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Is this a justified expression for the integral of the floor function?

Mathematica seems to agree with me in general with saying that $\displaystyle\int \lfloor x \rfloor dx = \frac{\lfloor x\rfloor (\lfloor x\rfloor-1)}{2}+\lfloor x\rfloor \{ x \}+C = \frac{\lfloor x\...
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0answers
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Correlation between the weak solutions of a differential equation and implied differential equations

Yes,this is very similar to a previous question I asked. That was about normal solutions and not weak solutions. We define the operator known as the implied derivative denoted as $I(f)(x)(g)$ to be: ...
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Product of integral and fractional part of binomial expansion

Let $R = (5\sqrt{5} + 11)^{2n+1} = [R] + f$, where $[.]$ denotes the greatest integer function. Prove that $Rf = 4^{2n+1}$. I need help with the above question. When I try expanding, I will get ...
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$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
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1answer
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Generalization of $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$

I have been asking the following question at MSE with an answer: $\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true? I found this relational ...
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Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int\left \lfloor{x}\right \rfloor dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left \lfloor{x^2+x-1}\right \...
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How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?

This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this. ...
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Solutions to $\frac1{\lfloor x\rfloor}+\frac1{\lfloor 2x\rfloor}=\{x\}+\frac13$

Find all solutions to $$\dfrac{1}{\lfloor x\rfloor}+\dfrac{1}{\lfloor 2x\rfloor}=\{x\}+\dfrac{1}{3}$$ $$$$ Unfortunately I have no idea as to how to go about this. On rearranging, I got $$3\lfloor 2x\...
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How to prove that $a<S_n-[S_n]<b$ infinitely often

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that $...
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Integer part of a sum (floor)

Let $\left(\, x_{n}\,\right)_{\,n\ \geq\ 1}$ be a sequence defined as follows: $$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$ Compute the ...
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1answer
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Iterating through the jump discontinuities of $f(x)$ [closed]

What I am attempting to do is the following. I want to find for some given function $f$ the unique step function $c$ (up to an arbitrary constant) such that $f(x) - c(x)$ is continuous assuming it ...
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prove that $\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$

How can I prove that if $x$ and $y$ are positive then $$\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$$
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Prove the following sequence always results in a perfect square. [duplicate]

There's a problem statement: For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} ...
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Analyzing $\biggl\lfloor{\frac{x}{5}}\bigg\rfloor=\bigg\lfloor{\frac{x}{7}}\bigg\rfloor$

How many non negative integral values of $x$ satisfy the equation :$$\biggl\lfloor{\dfrac{x}{5}}\bigg\rfloor=\bigg\lfloor{\dfrac{x}{7}}\bigg\rfloor$$. My try: Writing few numbers and putting in the ...
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3answers
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Challenging $\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}$ for $\epsilon=\frac{1}{2}$.

Consider the (incorrect) claim that $$\lim_{x \rightarrow 10} \frac{1}{\lfloor x \rfloor} = \frac{1}{10}.$$ How might I find the largest $\delta$ such that I can challenge $\epsilon = 1/2$? Clearly ...
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2answers
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Continuous antiderivative of $\frac{1}{1+\cos^2 x}$ without the floor function.

By letting $u = 2x$ and $t = \tan \frac{u}{2}$, I found the continuous antiderivative of the function to be: $$\int \frac{1}{1+\cos^2 x}dx\\= \int \frac{2}{3+\cos2x} dx\\ = \int \frac{1}{3+\cos u}du \...
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2answers
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Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'...
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1answer
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Conjectures Involving floor(x)/piecewise continuity with regards to Integration and Differential Equations

[This answer has been heavily edited in response to a long chain of comments on Eric Stucky's answer.] I came up with these few theorems and I am curious whether or not my hypothesis are true. I don'...
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2answers
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Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
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1answer
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Proof about floor function: $\lfloor x\rfloor+\lfloor x + \frac{1}{n} \rfloor+\cdots + \lfloor x + \frac{(n-1)}{n} \rfloor = \lfloor nx \rfloor$ [duplicate]

How can we prove that $$\left\lfloor x\right\rfloor + \left\lfloor x + \frac{1}{n} \right\rfloor + \left\lfloor x + \frac{2}{n} \right\rfloor + \cdots + \left \lfloor x + \frac{(n-1)}{n} \right\rfloor ...
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0answers
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Bounds on floor summation $\sum^n_k \lfloor c \cdot k \rfloor $

I've seen a couple posts recently about finding closed forms of or calulating $ \sum^n_k \lfloor c \cdot k \rfloor $ for any real $c>0$. And I was wondering about bounds we can give to this ...
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2answers
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Difference of the floor of a product and the product of floors

Is there any way the following can be simplified? $$\lfloor f(x)\cdot g(x) \rfloor - \lfloor f(x) \rfloor \cdot \lfloor g(x) \rfloor$$