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Questions tagged [ceiling-and-floor-functions]

This tag is for questions involving the greatest integer function (or the floor function) and the least integer function (or the ceiling function).

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Limit of an Integral with a Floor Function

I'm looking to evaluate the following integral and would like to verify my answer: $$\lim_{n\to\infty} \frac{1}{n^2} \int_{0}^{n^2} \sqrt{\frac{n^2-\lfloor \sqrt{x}\rfloor^2}{x}} \ \textrm{d}x$$ I've ...
Stamp's user avatar
  • 399
1 vote
2 answers
47 views

Confused on answer given for the question: prove $\lceil\frac{n}{m}\rceil = \lfloor\frac{n+m-1}{m}\rfloor$

1https://math.stackexchange.com/a/1281757/801877 Surprisingly, when trying to prove this problem I tried doing this exact way and got piecewise definition for ceiling(n/m), but I'm not sure how to do ...
Bob Marley's user avatar
0 votes
1 answer
48 views

Trying to see if $\lfloor x - y\rfloor = \lfloor x\rfloor - \lfloor y\rfloor$ [duplicate]

So let $\lfloor x\rfloor = n,\ \lfloor y\rfloor = m$ Then by definition we know that: $ n \leq x < n + 1$ and $ m \leq y < m + 1$ But when I subtract the 2 inequalities together, I get $n - m \...
Bob Marley's user avatar
1 vote
1 answer
57 views

Confusion on floor and ceiling function proof question.

These are questions from the Discrete Math textbook by Rosen, and I was just confused on what they're exactly asking: specifically when they write "... , when it is the larger/smaller of 2 ...
Bob Marley's user avatar
4 votes
0 answers
26 views

Does half of the sequence $\left(n^{\alpha}\right)_{n=1}^{\infty}$ have even integer part/floor?

Define $e\left((a_n)_{n=1}^{\infty};N\right)$ to be the amount of members of the sequence $(a_n)_{n=1}^{\infty}$ that are $\leq N$ and have even integer part (also known as the floor of the number). ...
Adam Rubinson's user avatar
0 votes
1 answer
47 views

The sum of Floor function

I am looking of calculating the following Sum. $\sum\limits_{k=0}^m \lfloor n/2^k \rfloor $ when $l$ is an integer and satisfies $n=2^m+l, 0\le l <2^m$ So far I’ve got $\sum\limits_{k=0}^m \lfloor ...
Russel0201's user avatar
0 votes
1 answer
49 views

Floor function and Golden ratio

Consider implicit equation : $$x=\left(\left(y^{-⌊x⌋}+x\right)^{-\frac{1}{⌊x⌋}}-y\right)\left(\sqrt{y^{-⌊x⌋}+x}+y\right)$$ I have observed that $$\text{when} \ x \to 0^{-}, \ y \to \frac{1}{\Phi^{2}}$$...
Pin Pin's user avatar
  • 11
2 votes
4 answers
106 views

$\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]=k-1$

Consider this $\left[\dfrac 12\displaystyle\sum_{n=1}^{k^2}\frac 1{\sqrt n}\right]$ where $[\cdot]$ is the greatest integer function. I had observed that its value is $(k-1)$ by putting $k=2,3,4,$ ...
user1318878's user avatar
4 votes
1 answer
104 views

Accumulation points of the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor$

I'm trying to find the accumulation points for the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor, n \in \mathbb{N}_0$. I know what the points are, but I'm having trouble coming up with an explicit ...
Focus's user avatar
  • 43
4 votes
1 answer
222 views

Solve for $x$: $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=2001$

Solve for $x$: $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=2001$ Given solution: $\lfloor x\rfloor=\lfloor \sqrt[4]{2001}\rfloor=\{6,-7\}$ If $\lfloor x\rfloor=6\Rightarrow x\lfloor x\lfloor 6x\...
Maverick's user avatar
  • 9,463
2 votes
1 answer
93 views

Solve for $x$: $\lfloor 2^x\rfloor+\lfloor 3^x\rfloor=\lfloor 6^x\rfloor$

Solve for $x$: $\lfloor 2^x\rfloor+\lfloor 3^x\rfloor=\lfloor 6^x\rfloor$ My Attempt: Clearly the equation is satisfied for all $x<0$. For positive $x$, $2^x+3^x-6^x=\{2^x\}+\{3^x\}-\{6^x\}$ $-1<...
Maverick's user avatar
  • 9,463
11 votes
2 answers
355 views

What is $\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$?

I came up with this while messing around with the $\gcd$ and $\operatorname{lcm}$ functions in Desmos. $$I(a)=\int_{0}^{\pi/2}\frac{\operatorname{lcm}(a\cos x,a\sin x)}{a^2}dx$$ The function inside ...
Dylan Levine's user avatar
  • 1,666
0 votes
0 answers
36 views

Challenge: Define rounding functions using integer checking functions

This question was inspired by Willans' formula, in which he used the cosine function as a way to constrain a number and floor it to be able to count primes (see https://www.youtube.com/watch?v=...
Yusuf Maher's user avatar
2 votes
0 answers
28 views

Finding the sum of the floor function of $a,(b-1)/2,c$ given two symmetric sums

Problem: Let $a<b<c$ be $3$ real numbers satisfying $a+b+c=6$, $ab+bc+ca=9$. Then, determine the value of $\lfloor{a}\rfloor+\lfloor\frac{b-1}{2}\rfloor+\lfloor{c}\rfloor$. My method of solution:...
Cognoscenti's user avatar
2 votes
3 answers
87 views

Calculate the whole part of $A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$

the question Calculate the whole part of: $$A=\frac{1012^2}{\sqrt{1*2}+\sqrt{3*4}+\cdots+\sqrt{2023*2024}}$$ my idea My first thought was to to fit the number between 2 consecutive natural numbers ...
IONELA BUCIU's user avatar
1 vote
0 answers
28 views

Looking for a solution to a 2-dimensional recurrence relation with a floor function in GF(2)

I have a recurrence relation that i'd like to solve. It is defined as follows: $$ a_{i,j} = (a_{i-1,j} + \lfloor 3 \sum_{k=0}^{j-1} \frac{a_{i-1,k}}{2^{j-k}} \rfloor \space) \bmod 2 $$ where $ a_{i,...
BlueLisztmaths's user avatar
0 votes
1 answer
16 views

Y~Geometric(p) then why does the CDF of Y have the floor function?

Here is the screenshot from my textbook, Hossein Pishro-Nik Here, I have calculated the CDF of Y to be 1 - q^y. But the next step involved y being a floor function. I am not sure why that is the case ...
Arnav Aditya's user avatar
1 vote
0 answers
51 views

Positive solutions to a linear Diophantine equation

Let $d,d',n\in \mathbb N$ be given. If you want, assume $(d,d')=1$. How many positive integer solutions does $$dx+d'x'=n$$ have? (Assuming $(d,d')=1$). I know there are $n/dd'+\mathcal O(1)$ solutions,...
tomos's user avatar
  • 1,732
0 votes
0 answers
65 views

Weird Property of Irrational Numbers and their "Conjugate"

Let us have a irrational number $a$. Let $a'$ be $\frac{1}{a'} + \frac{1}{a} = 1$. If we let $A$ and $B$ be sets such that $A =$ {$\lfloor a \rfloor, \lfloor 2a \rfloor, \lfloor 3a \rfloor, \lfloor 4a ...
bob john's user avatar
0 votes
0 answers
27 views

Proving the Greatest Integer Fucntion (G.I.F) of (50!/e + 1/2) is D50(derangement)

Prove that, If D(n) denotes dearrangement for n objects, then D(50) = [ 50/e! + 1/2 ] Where [.] denotes the greatest integer function. Through exapnsion i generated 50/e! but how can we adjust the &...
OpateItZOpatoOpate's user avatar
0 votes
0 answers
20 views

Points of discontinuity of greatest Integer function

So here's my problem Find the points of discontinuity of $[x^2]$ in the interval$[-1,2]$ where [] is the G.I.F. I am not able to understand why at $2^{1/2}$ and $3^{1/2}$ the function is being ...
πααρτθ Σαρθι's user avatar
1 vote
1 answer
20 views

Quesiton regarding nature of equations involving fractional part and greatest integer functions

This question is in context of the following problem Solve: $[x]^2 = x + 2\{x\}$ Where $[.]$ and $\{.\}$ represent the greatest integer and fractional part function respectively. The solution for the ...
koiboi's user avatar
  • 736
3 votes
2 answers
35 views

Question regarding intersection of values of a function

This question is being asked in regards to the following problem Solve: $x^2 - 4 - \lfloor x \rfloor = 0$ The solution for the above equation in my textbook is done by drawing the graph of both the ...
koiboi's user avatar
  • 736
1 vote
2 answers
37 views

Question regarding property of greatest integer function

In my textbooks section about the properties of the greatest integer function the two following properties looked quite interesting and i could not understand them right away. Property 1 $$\lfloor(\...
koiboi's user avatar
  • 736
0 votes
1 answer
60 views

Computing $h(h(x))$ where $h (x) = \lfloor 5x - 2 \rfloor$

In Velleman's "Calculus: a Rigorous Course," Example 9 from Section 1.3 tasks us with computing $ h(h(x)) $, where $ h(x) = \lfloor 5x - 2 \rfloor $. My initial solution: \begin{align*} h(\...
F. Zer's user avatar
  • 2,357
1 vote
2 answers
67 views

How to compute the following limit? $\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$ [closed]

$$\lim\limits_{x\to\ 0^+} {\frac{x-\lfloor x \rfloor}{x+\lfloor x \rfloor}}$$ Here, $\lfloor x \rfloor$ represents the floor of $x$. I tried using a graphing calculator (desmos) to plot the function $...
Jesko's user avatar
  • 45
4 votes
1 answer
164 views

Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$

ORIGINAL QUESTION (UPDATED): I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function: $$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \...
Richard Burke-Ward's user avatar
2 votes
3 answers
257 views

Prove⌈a/b⌉ ≤ a/b + (b-1)/b [closed]

For integers $a, b > 0$, Prove $⌈a/b⌉ ≤ (a + (b-1))/b$ RHS $= a/b + (b-1)/b $ where $ (b-1)/b $ is $[0,1)$ If $a/b$ is an integer, inequality holds true as we are adding non-negative term. If $a/b$ ...
jam's user avatar
  • 61
0 votes
1 answer
81 views

For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy\ldots$

For $0<x<1$, let $f(x)=\int_0^1\left( \left\lfloor\frac{x}{y}\right\rfloor-x \left\lfloor\frac{1}{y}\right\rfloor\right)dy$, $\lfloor. \rfloor$ denotes the greatest integer function. If $f(x)$ ...
mathophile's user avatar
  • 3,753
2 votes
1 answer
34 views

Asymptotics of expression involving floor functions

Consider two functions $f_1(n)$ and $f_2(n)$ that grow to infinity at the same speed, say $$\lim_{n\to \infty} \frac{f_1(n)}{f_2(n)}=c $$ for some $c>0$. I am studying the squared difference of $...
FishPie's user avatar
  • 43
1 vote
2 answers
99 views

$f(n)= \left\lfloor \frac{an+b}{cn+d} \right\rfloor , \forall n \in \mathbb N$ is surjective

Let $a,b,c,d \in \mathbb N , d \ne 0$ and consider the funtion $f: \mathbb N \rightarrow \mathbb N $ such that : $$f(n)=\left \lfloor \frac{an+b}{cn+d}\right \rfloor , \forall n \in \mathbb N$$ Prove ...
Unknowduck's user avatar
2 votes
4 answers
125 views

Graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$

$\lim_{x\to0+}(\lfloor x \rfloor + \lfloor1-x\rfloor)$ $\lim_{x\to0-}(\lfloor x \rfloor + \lfloor1-x\rfloor)$ I tried to solve by graphing $(\lfloor x \rfloor + \lfloor1-x\rfloor)$ Graph of the ...
Ak9848's user avatar
  • 35
2 votes
3 answers
96 views

Solving equation with two floor functions

I'm trying to solve the following question \begin{equation*} \left\lfloor \frac{\left\lfloor \frac{3\lfloor x\rfloor }{2}\right\rfloor }{9}\right\rfloor =4 \end{equation*} I got the following ...
dagoda's user avatar
  • 31
2 votes
0 answers
31 views

Solve recurrence $A(0)=0$, $A(n)=1+\min_{1\leq k\leq n}\left\{\max\left(\left\lfloor\frac{k-1}{2}\right\rfloor, A(n-k)\right)\right\}$.

I am asked to solve the following recurrence problem: $$A(0)=0,\quad A(n)=1+\min_{1\leq k\leq n}\left\{\max\left(\left\lfloor\frac{k-1}{2}\right\rfloor, A(n-k)\right)\right\}\,. $$ I have observed ...
YanRuiJie's user avatar
0 votes
1 answer
47 views

integration of floor function [closed]

How to Integrate the following? $$\int_{2}^{343}\{x-\lfloor{x}\rfloor\}^2dx = \int_{2}^{3}x^2dx + \int_{3}^{4}x^2dx+\cdots+\int_{342}^{343}x^2dx$$ This is what I did, How can I proceed further with ...
econhead's user avatar
0 votes
0 answers
17 views

How to reduce division with uprounding to integer-arithmetic operations with downrounding if the denominator is not necessarily an integer?

Let 𝑖∈ℕ₀ be an unknown variable and 𝑐∈ℚ₊ be a known constant such that 𝑐≥2 and 100𝑐 ∈ ℕ₊. (So we can pre-compute anything regarding 𝑐, e.g., represent 100𝑐 as a product of powers of primes.) We ...
AlMa1r's user avatar
  • 101
3 votes
5 answers
113 views

Solve $\lfloor x-1\rfloor(3^x-2^x-\lfloor x^2\rfloor) = 0$

Solve the following equation in $\mathbb R$ $$\lfloor x-1\rfloor(3^x-2^x-\lfloor x^2\rfloor) = 0$$ where $\lfloor y\rfloor=k \iff k \le y < k+1 , k \in \mathbb Z$ I will address 2 cases. $\lfloor ...
Unknowduck's user avatar
0 votes
0 answers
46 views

Inverse of a piecewise floor function

Given the following definitions $$ y(x) = \left\lfloor{ (x - n + a - b)/a }\right\rfloor ,\quad a,b \in \mathbb{R}, n \in \mathbb{N}, $$ $$ z(x) = \max (0, y(x)), $$ where $x, y(x), z(x) \in \mathbb{N}...
Toool's user avatar
  • 307
1 vote
2 answers
78 views

Simplifying $\sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$

how to simplify this expression so that I get an expression only? when $t = 1, 2,.... $ $$y = \sum _{t=1}^{\infty }\:\frac{\left(1+g\right)^{ \lceil t/2\rceil}}{\left(1+k\right)^t}$$ I know that for t ...
CountDOOKU's user avatar
  • 1,041
0 votes
2 answers
45 views

Inequality for integers with floor function

I want to show that for any nonnegative integers $l$ and $b$ we have $$ \frac{l}{2^{b+1}} - 1 \leq \left\lfloor \frac{l-1}{2^{b+1}} \right\rfloor. $$ I have a proof where I wrote $l = \alpha\cdot 2^{...
Lereu's user avatar
  • 434
2 votes
3 answers
73 views

Solve in $\mathbb{R}^+$ algebra calculus Equation with derivatives

I was trying to solve this question: Solve in $\mathbb{R}^+$ the Equation $7^x-5^x=\lfloor x^2 \rfloor+1$ Where $\lfloor t\rfloor$ is the floor function It is known that $\lfloor x^2\rfloor\leq x^2 $ ...
Kolsinki Vetko's user avatar
0 votes
1 answer
41 views

Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ is $1 \frac{1}{3}$?

Prove that the maximum difference between $\frac{n}{6}$ and $\lfloor \frac{(n-3)}{6} \rfloor$ where $n$ is an integer is $1 \frac{1}{3}$? I know this is true because for $n = 8 + 6i$ where $i$ is an ...
Kenneth Watanabe's user avatar
1 vote
1 answer
92 views

Is there a nice form to $\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$?

$$\sum_{k=1}^{n}\left\lfloor\sqrt{ r^2-k^2 }\right\rfloor$$ where $r$ is a constant (not necessarily an integer). Note that $r\ge n$ and both $r, n$ are positive. I apologize if I'm not able to ...
Lucien Jaccon's user avatar
5 votes
1 answer
182 views

Integer part of $x^n$

Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$? For simplicity, let us assume $x<2$. For the first few values of $n$, ...
Dumbest person on earth's user avatar
1 vote
1 answer
81 views

Limit involving an Indicator Function and Sum

For each $n\in\mathbb{N}$, consider real $n$ numbers $x_{i,n}$, $1\le i \le n$, given by $$\sum_{i=1}^{n}x_{i,n}=0,\\-1/n \le x_{i,n} \le 1-1/n.$$ I am trying to find value of $$\lim_{n\to \infty} \...
The Substitute's user avatar
3 votes
2 answers
206 views

Integral containing floor function and derivative

Let $n>1$ be a positive integer, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be continuously differentiable on the interval $[1,n]$. I want to calculate an integral of the form $$ \int _1^n\...
Richard Burke-Ward's user avatar
2 votes
2 answers
103 views

Is there a spelling mistake or am I missing something

Here, $[ \cdot]$ is $\lfloor \cdot \rfloor $ floor function. N $ \in N$. Where did $\frac{[Nx]} N + \frac{1}{2N}$ came from and how does $x$ differs by $\frac{1}{2N}$. Shouldn't it be $\frac{1}N$ if ...
Yugant Shewale's user avatar
1 vote
3 answers
128 views

Find the minimum value of $(x-y)^2+(x-y+ \frac{1}{y}-\frac{1}{x})^2$ where $x>0>y$

If $\lambda$ denotes the minimum value of $(x-y)^2+(x-y+ \frac{1}{y}-\frac{1}{x})^2$ where $x>0>y$, then find the value of [3$\lambda$] where [.] denotes the greatest integer function. My ...
Rexquiem's user avatar
  • 334
1 vote
1 answer
55 views

What is the 1-case closed form for $\sum_{i = 1}^{x} \lfloor \frac{i - r}{d}\rfloor$?

Let all untyped variables be natural numbers. Formula? Given $x \geq 1$, $0 \leq r \lt d$ there are two cases to handle: $x \lt r$ and $x \geq r$. What is the 2-case closed form for $\sum_{i = 1}^{x} \...
Daniel Donnelly's user avatar
0 votes
1 answer
46 views

approximate a sum [closed]

Is there a way to simplify the given function: $f(n):={\sum}_{x={\lceil\frac{n}{\mathrm e}\rceil}}^{n}\frac{\ln\left(\frac{n}{x+1}\right) \left(x+1\right)}{\left(x-1\right) x}$, with $n>...
TooMath's user avatar
  • 35

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