# 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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### 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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### 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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### 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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### 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 ...
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) $\int_0^2$ $\left \lfloor{x^2+x-1}\right \... 4answers 2k views ### How to show$\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? This question came from the prelim exam I took last month. I have a proof that seems a bit unwieldy to me (posted as an answer), so I'm opening it up to ask if there are other ways of showing this. ... 4answers 206 views ### Solutions to$\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 $$3\lfloor 2x\... 4answers 330 views ### How to prove that a<S_n-[S_n]<b infinitely often Let S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}, where n is a positive integer. Prove that for any real numbers a,b,0\le a\le b\le 1, there exist infinite many n\in\mathbb{N} such that ... 3answers 700 views ### Integer part of a sum (floor) Let \left(\, x_{n}\,\right)_{\,n\ \geq\ 1} be a sequence defined as follows:$$ x_{1}={1 \over 2014}\quad\mbox{and}\quad x_{n + 1}=x_{n} + x_{n}^{2}\,, \qquad\forall\ n\ \geq\ 1 $$Compute the ... 1answer 246 views ### Iterating through the jump discontinuities of f(x) [closed] What I am attempting to do is the following. I want to find for some given function f the unique step function c (up to an arbitrary constant) such that f(x) - c(x) is continuous assuming it ... 6answers 310 views ### prove that \lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor How can I prove that if x and y are positive then$$\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$$2answers 395 views ### Prove the following sequence always results in a perfect square. [duplicate] There's a problem statement: For each m \in \mathbb{N}, we construct a sequence m_0, m_1, m_2,\dots denoted S_m, recursively via m_0=m and$$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} ... 4answers 179 views ### 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 ... 3answers 862 views ### 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 ... 2answers 232 views ### 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: $$\int \frac{1}{1+\cos^2 x}dx\\= \int \frac{2}{3+\cos2x} dx\\ = \int \frac{1}{3+\cos u}du \... 2answers 328 views ### Analytic floor function, why this seems to work? I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'... 1answer 486 views ### Conjectures Involving floor(x)/piecewise continuity with regards to Integration and Differential Equations [This answer has been heavily edited in response to a long chain of comments on Eric Stucky's answer.] I came up with these few theorems and I am curious whether or not my hypothesis are true. I don'... 2answers 102 views ### Prove \Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor , n is a Natural Number Prove the following identity:$$\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$$... 1answer 602 views ### Proof about floor function: \lfloor x\rfloor+\lfloor x + \frac{1}{n} \rfloor+\cdots + \lfloor x + \frac{(n-1)}{n} \rfloor = \lfloor nx \rfloor [duplicate] How can we prove that$$\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 ... 0answers 502 views ### 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 ...
Is there any way the following can be simplified? $$\lfloor f(x)\cdot g(x) \rfloor - \lfloor f(x) \rfloor \cdot \lfloor g(x) \rfloor$$