Question

Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$

score 519 · Answer 1 · 2010-09-28 20:14:32

519

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Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$

Published on 28 Sep 2010 - 20:14

score 3.8k · Answer 2 · 2010-10-05 06:50:02

3.8k

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For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

Published on 05 Oct 2010 - 6:50

score 4.5k · Answer 3 · 2026-03-31 08:45:14

4.5k

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Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Published on 31 Mar 2026 - 8:45

score 3.2k · Answer 4 · 2010-11-13 22:54:35

3.2k

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Floor of Square Root Summation problem

Published on 13 Nov 2010 - 22:54

score 493 · Answer 5 · 2011-03-17 13:04:48

493

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Algebraic notation of floor function in an equation

Published on 17 Mar 2011 - 13:04

score 3.9k · Answer 6 · 2011-03-31 20:01:17

3.9k

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How to solve these equations that involve the ceiling function

Published on 31 Mar 2011 - 20:01

score 6.9k · Answer 7 · 2011-04-17 21:36:25

6.9k

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Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

Published on 17 Apr 2011 - 21:36

score 1.4k · Answer 8 · 2011-04-17 22:58:55

1.4k

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Is this a justified expression for the integral of the floor function?

Published on 17 Apr 2011 - 22:58

score 2.4k · Answer 9 · 2011-05-07 05:02:02

2.4k

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How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$?

Published on 07 May 2011 - 5:02

score 4.4k · Answer 10 · 2011-05-28 01:02:54

4.4k

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How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?

Published on 28 May 2011 - 1:02

score 760 · Answer 11 · 2011-07-02 07:23:35

760

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The floor of $\sqrt{2x}+1/2$ is the ceiling of $(\sqrt{1+8x}-1)/2$

Published on 02 Jul 2011 - 7:23

score 443 · Answer 12 · 2011-08-29 05:34:23

443

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$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

Published on 29 Aug 2011 - 5:34

score 1.4k · Answer 13 · 2011-09-10 13:30:03

1.4k

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Integral and derivative of $\lfloor x\rfloor$ and $x - \lfloor x\rfloor$

Published on 10 Sep 2011 - 13:30

score 398 · Answer 14 · 2011-12-03 08:58:10

398

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Convergence of limit with integer part of $x$: $\lim_{x\to 0^+}x^2\left(\Bigr[\frac1x\Bigr]+\Bigr[\frac2{x}\Bigr]+\dots+\Bigr[\frac{k}{x}\right)$

Published on 03 Dec 2011 - 8:58

score 2.7k · Answer 15 · 2026-03-26 18:59:41

2.7k

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$\lfloor (2+\sqrt{3})^n \rfloor $ is odd

Published on 26 Mar 2026 - 18:59

List Question

Proving $\sum\limits_{k=0}^{n-1} \Bigl[x + \frac{k}{n}\Bigr] = [nx]$

For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$

Floor of Square Root Summation problem

Algebraic notation of floor function in an equation

How to solve these equations that involve the ceiling function

Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$

Is this a justified expression for the integral of the floor function?

How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$?

How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$?

The floor of $\sqrt{2x}+1/2$ is the ceiling of $(\sqrt{1+8x}-1)/2$

$2 \lfloor x \rfloor \leq \lfloor 2x \rfloor \leq 2 \lfloor x \rfloor +1$

Integral and derivative of $\lfloor x\rfloor$ and $x - \lfloor x\rfloor$

Convergence of limit with integer part of $x$: $\lim_{x\to 0^+}x^2\left(\Bigr[\frac1x\Bigr]+\Bigr[\frac2{x}\Bigr]+\dots+\Bigr[\frac{k}{x}\right)$

$\lfloor (2+\sqrt{3})^n \rfloor $ is odd

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