Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$

Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$ My Attempt: I took intervals of $x$ as $x\in (2,3)$ so $\sqrt x\in (1,2)$. Due to which $\lfloor x\rfloor=2$ and $\{\...
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  • 7,052
2 votes
1 answer
101 views
+50

Questions about density of $\left\{\ \left( k+ \frac{1}{2} \right)^n \right\}$ in $[0,1]$

Here, $\{ x \}$ denotes the fractional part of $x.$ Are there any known positive integers $k$ for which the set $\left\{\ \left\{ \left( k+ \frac{1}{2} \right)^n \right\}: n\in\mathbb{N}\ \right\} $ ...
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1 vote
2 answers
24 views

Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.)

Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.) My Attempt: Replacing $x$ by $[x]+\{x\}$, where $\{x\}$ is a fractional ...
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  • 5,236
4 votes
1 answer
64 views

Fractional part and greatest integer function

Here are a few questions on the fractional part and the greatest integer function. Find out $[\sqrt[3]{2022^2}-12\sqrt[3]{2022}]$ If $\{x\}=x-[x],$ find out $[255\cdot x\{x\}]$ for $x=\sqrt[3]{15015}....
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  • 1,266
-1 votes
0 answers
52 views

Show that the limit of $\{n\theta_n\}$ goes to $\frac{1}{4}$. [closed]

Let the function $f$ be four times continuously differentiable on $[-1,1]$ with $f^{(4)}(0)\neq 0$. For each $n\geq 1$, let $$f\left(\frac{1}{n}\right)=f(0)+\frac{1}{n}f'(0)+\frac{1}{2n^2}f''(0)+\frac{...
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2 votes
1 answer
45 views

Given irrational numbers $0<\gamma_1<\gamma_2<1,$ real numbers $0\leq a<b\leq 1,$ does $\exists n$ such that $\{n\gamma_1\}, \{n\gamma_2\}\in(a,b)?$

For this question, $\{x\}$ means the fractional part of the real number $x.$ Given irrational numbers $0<\gamma_1<\gamma_2<1,\ $ real numbers $a,b$ with $0\leq a<b\leq 1,$ does there exist ...
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-1 votes
0 answers
27 views

Can we say multiplication of fraction means part of wholes of fractions

Can we say multiplication of fractions means part of the wholes or fractions? For example, 7/4 × 3 means 7/4 of 3. Does it mean ...
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  • 49
0 votes
1 answer
38 views

Are the [x] and {x} functions defined for negative numbers? [duplicate]

[x] is the whole part (floor) of x {x} is the fractional (decimal) part of x ...
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0 votes
0 answers
30 views

Integral of factional part

Given a function: $$ F(k) = \sum_{k=600}^{999} \left(\left(\pi^2 \right(\frac{k}{1000} + x - 1\left)^2 \right)^{10^{-7}}\right)$$ where $0.1<x<0.35$. The remainder term of the Euler Maclaurin ...
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1 vote
0 answers
67 views

prove that $ \int_0^{\infty}\frac{\{x\}\cos x}{x} dx \neq \infty $ [closed]

I can't prove that $ \int_0^{\infty}\frac{\{x\}\cos x}{x} dx \neq \infty $ ({x} — fractional part x). Calculations show that this integral is approximately 0.211... It suffices for me to prove that ...
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3 votes
1 answer
148 views

Find the sum/asymptotic as $N \rightarrow \infty$ of $\sum_{a = a_1}^{a_2} \big\{{\frac{{a}^{2}-N}{2\, a}}\big\}$

Find the sum (if possible) and asymptotic expansion (necessary) to as many terms as possible as $N \rightarrow \infty$ for $$ S \left({N}\right) = \sum_{a = \lceil\sqrt{N + 2}\rceil}^{\lfloor 2N/3 \...
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0 votes
1 answer
46 views

Studying convergence of the series involving fractional part

I need to study where the following series is convergent: $$\displaystyle \sum_{n=1}^{\infty} \frac{\{ n \alpha\}}{n}$$ Here $\{x\}$ denotes the fractional part of $x$ and $\alpha \in \mathbb{R}_{>...
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5 votes
1 answer
148 views

The limit $\lim_{n \to \infty} \int_0^1 f(\{n x\}) g(\{n/x\}) \, \mathrm{d} x$

This is problem 1.82 in Ovidiu Furdui's book 'Limits, Series, and Fractional Part Integrals'. Given $f,g \in \mathrm{C}([0,1])$, let $$ a_n = \int \limits_0^1 f(\{n x\}) g\left(\left\{\frac{n}{x}\...
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0 votes
0 answers
34 views

Fractional part sequence

I am having one question that ask finding one sequence defined by $b(n) = \{a_n\} = a_n - [a_n]$ , where $\{x\} = x - [x]$ is a fractional part of a real number $x$ and $[x]$ is the largest integer ...
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  • 21
2 votes
3 answers
105 views

How do we solve $x^2 + \{x\}^2 = 33$ without computer?

This is a problem taken from a group on Facebook. I wonder how to solve this without numerical process. $x^2 + \{x\}^2 = 33\tag{1}$ My unfinished attempt: $$\begin{align} x^2 + \{x\}^2 &= 33\\ x^2 ...
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  • 2,075
9 votes
3 answers
177 views

$\{x^2\} = \{x\}^2$, how many solutions in interval $[1, 10]$

Find how many solutions there are in the interval $[1, 10]$ to the fractional part equation: $$\left\{x\right\}^2 = \left\{x^2\right\}$$ Where $\{\cdot\}$ is the fractional part function, meaning that:...
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0 votes
2 answers
66 views

If $\{x\}$ denotes the fractional part of x, then what is $\{\dfrac{3^{2n}}{8}\}$, where n $\in$ N

I obtained this question from my school provided material. At first, it's quite easy to see through simple substitution that for any natural value of $n$, we get the answer as $\dfrac{1}{8}$, but I ...
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1 vote
1 answer
65 views

$\{a\}+\{1/a\}=1$ prove that $\{a^3\}+\{1/a^3\}=1$

If $a$ is a real number such that If $\{a\}+\{1/a\}=1$ prove that $\{a^3\}+\{1/a^3\}=1$, where $\{x\}=x-[x]$ for any $x$ real number I got that $a+1/a-[a]-[1/a]=1$ but I do not think this is useful ...
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2 votes
0 answers
86 views

Number theory Question From RMO(2015) regarding fractional part function. [duplicate]

Show that there are infinitely many positive real numbers $a$ which are not integers such that $a(a-3\{a\})$ is an integer (here $\{a\}$ denotes the fractional part of $a$). I tried putting $a = [a]+\{...
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0 votes
1 answer
37 views

Better way to compute this definite piecewise integral

I have to find the value of $$I = \int_0 ^{10} [x]^3\{x\}dx$$ Where $[x]= $greatest integer less than or equal to$ x $(the greatest integer function or the floor function) And $\{x\}= $fractional part ...
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  • 1,370
1 vote
2 answers
140 views

Prove that $(3/2)^n\bmod 1\equiv (2^n-1)/2^n $ has infinitely many solutions as $n \to\infty$ [closed]

The first part of the question is false and was proved by the answer vefore. The second part of the question asks to prove that $(3/2)^n \bmod 1\equiv (2^n-1)/2^n$ also has infinite solutions as $n \...
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3 votes
1 answer
245 views

Summation of square root fractional part

Is there a way to calculate the following sum for large $c>K$, closed-form or approximation? $$\sum_{n=1}^{c}\left\{\sqrt{K+n^2}\right\}$$Where K and n are positive integers and $\{\}$ indicates ...
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2 votes
2 answers
88 views

Domain of $f(x)=1/\sqrt{\left\{x+1\right\}-x^2+2x}$ where {.} denotes the fractional part of $x$.

Normally, in finding domain for square root function, what I do is that what ever is under the root should be equal to or greater than $0$. Here the square root is in denominator so $\left\{x+1\right\}...
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0 votes
0 answers
30 views

What is the intuitive meaning of mixed partial derivative of the following fractional function?

I have a situation to use the following form of 2nd order mixed partial derivative. So I tried to manipulate it into intuitive form but still don't understand its meaning. $\frac{\partial^2}{\partial ...
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  • 1
0 votes
0 answers
25 views

Dividing an interval into sections (fractional)

I found this question in an examination, questions like these are barely covered in my unit, so I'm quite stumped as to how to address them. This was a two-part question, and the first asked to show $\...
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  • 1
2 votes
1 answer
141 views

How can I prove that $a_n>0$ infinitely often?

Let $n\in\mathbb{N}$ and $$a_n:=sin(2\pi^2(2n+1)!)$$ How can I prove that $a_n>0$ infinitely often? Clearly, $a_n>0$ infinitely often is equivalent to {$\pi(2n+1)!$}$\leq 0.5$ infinitely often ...
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2 votes
0 answers
81 views

Average Number of Small Divisors

I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
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2 votes
1 answer
68 views

Evaluating a Double Integral Involving Fractional Part Functions

In my project, I am stuck with the following integral. Please help. $$ I = \int_{x = a}^b \int_{y = a}^b \left\{ \frac xy\right\} \left\{\frac >yx\right\}~dx~dy$$ $0 < a < b < 1, b < ...
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0 votes
2 answers
87 views

Find the solution of $x^2-4=[x]$

I am able to find the solution by using the help of graph. I know $x^2-4$ will cut $[x]$ only at $-2$ and $2$ and then I am able to find the answer. I want to know, can we approach this question in ...
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0 votes
0 answers
39 views

Infinitely recursive summation of floor function

I'm actually dealing with the fractional part function, $$\{x\} = x - \lfloor x\rfloor$$ On Wikipedia, a Fourier series expansion is given as, $$\{x\} = \frac{1}{2} - \frac{1}{\pi}\sum_{k=1}^\infty \...
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1 vote
1 answer
39 views

Does $\lim_{k\to\infty}2^k \log_2(2^k-1)\equiv\gamma\mod 1$ where $k\in\mathbb{N}$ (Euler-Mascheroni Constant)

I was working with the function $f(k)=2^k \log_2(2^k-1)$ and had noticed that it's factional component seemed to converge when $k\in\mathbb{N}$. For example, $f(13)\equiv 0.557216896821\mod 1$, ...
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  • 3,726
1 vote
3 answers
107 views

Finding all rational values of $x$ for which $256(x + 15\{x\})\{x\} = 2021$

Find all rational numbers $x$ that satisfy $$256(x + 15\{x\})\{x\} = 2021$$ where $\{x\}$ is the fractional part of $x.$ I first substituted in $x - \lfloor x \rfloor = \{x\},$ which gave me $$256(...
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1 vote
1 answer
95 views

Solve ${x}^{\lfloor x \rfloor} = a$

I have found this weird equation in a math book. Could you give me any hints? $${x}^{\lfloor x \rfloor} = a$$ for a given a. I have dealt with the trivial cases where $x$ is an integer but cannot ...
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  • 909
6 votes
1 answer
219 views

Sum of two floors is equal

Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$,...
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  • 550
6 votes
2 answers
273 views

A possible upper bound on the fractional part of exponential functions

Does there exist a real number $a$ bigger than $1$ that's not a rational power of an integer such that $\displaystyle\lim_{n\to\infty}${$a^n$}$=0$ where values for $n$ are positive integers? (P.S. $\{ ...
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  • 1,411
0 votes
3 answers
66 views

{$\frac{1}{x}$} , $0<x\leq1$ in terms of {$x$}, $x\geq 1$, {$x$} is fractional part of $x$

$f(x)$ $=$ {$\frac{1}{x}$}, $0<x\leq1$ where {$x$} denotes the fractional part of $x$. $g(x) =$ {$x$}, $x\geq 1$ I want an expression for $f(x)$ in terms of x and $g(x)$. My try- If $x\in \mathbb{Z}...
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6 votes
1 answer
187 views

Find a closed form for the following integral:

I have to evalute a closed form for the following integral: $$\int_0^1\left\{\frac{(-1)^{\lfloor\frac{1}{x}\rfloor}}{x}\right\}dx$$ where $\{x\}$ is the fractional part. I thought of using a ...
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1 vote
0 answers
78 views

Apply the method of hyperpolas to $\sum_{n \le {x}^{1/k}} n * Fraction \left(\frac{x}{{n}^{k}}\right)$

I am using the example from Theorem 1 of Friedrich Pillichshammer "Euler's Constant and Averages of Fractional Parts" (https://www.dmg.tuwien.ac.at/nfn/gamma.pdf) we have for integers $k >...
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0 votes
1 answer
13 views

Simplification of complex rational?

I was looking at a problem in a textbook and found the following simplification: $$\frac{1}{x+i\omega} * \frac{2y}{y^2 + \omega^2} = \frac{2}{xy} * \frac{1}{1+i\frac{\omega}{x}}*\frac{1}{1 + (\frac{\...
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  • 201
2 votes
0 answers
22 views

How to visualize division as splitting Dividend into B equal "partial groups", then rounding up A partial groups to get a full group?

Because $10 \div \dfrac{4}{3}$ isn't an integer, I changed the numbers in this Reddit post. My $X = 6, A = 2, B = 3$. Undoubtedly, I know $6 \div \dfrac{2}{3} = 6 \times \dfrac{3}{2} = 9$, but don'...
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0 votes
1 answer
110 views

Series of integer part using fractionnal part fourrier series

I'm trying to calculate this sum: $F(N,p) = \sum_{k=1}^\infty \left\lfloor{\frac{2N+1}{2p^k}+\frac12+\frac{1}{\beta p^k}}\right\rfloor $ with $ N \in \mathbb N^*$, $ p \in \mathbb P$ and $\beta >2$ ...
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0 votes
2 answers
100 views

How to prove that frac($\sqrt{4^n-1}$) is greater than 1/2 for any natural n

I would like to prove that the fractional part of $\sqrt{4^n-1}$ is greater than 1/2 for any natural n. Just to clarify- $frac(x)...
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  • 605
2 votes
1 answer
155 views

Asymptotic estimate as $N \rightarrow \infty$ of $\sum\limits_{n = 1}^{N} \left\{{\frac{\left({n \pm 1}\right)}{{n}^{2}} N}\right\}$

Looking for the exact if possible, otherwise the asymptotic expansion and best estimate of the error terms as $N \rightarrow \infty$ of the two fractional sums $$\sum\limits_{n = 1}^{N} \left\{{\frac{\...
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3 votes
2 answers
311 views

Sum of fractional parts [closed]

So in analysis 1 we re doing limits of sums and general terms and so on and this is one of the exercises we have at homework. And just like most exercises that involve fractional parts, i dont even ...
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  • 308
1 vote
1 answer
163 views

Integral representation of $\zeta(s)$ in terms of the fractional part of $x$

I derived an integral representation of the Riemann zeta function, which is: $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\{x\}x^{-s-1}dx$$ where $\{x\}$ is the fractional part of $x$. Please verify my ...
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0 votes
4 answers
97 views

find out the fractional part for $\sqrt{25} + \sqrt{24}$ [closed]

I would like to know if there is any more elegant solution than the extraction of the root or the approximation. I tried something like this: $\sqrt{25}$ + $\sqrt{24}$ = $( \sqrt3 + \sqrt2 )^2$ But ...
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  • 311
1 vote
1 answer
42 views

For $a,b\in\mathbb{R}$, there is an integer within $|\{a\} - \{b\}|$ from $|a-b|.$

Let's take two real numbers $a,b$. The distance between $a$ and $b$ is $|a-b|$. Let $\{\}$ denote fractional part. Then for any $a$ and $b$, there is an integer close to $|a-b|$ which is at most $|\{...
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  • 221
1 vote
0 answers
134 views

Partial limits of $x_n = \{ \alpha n^2\}$

Consider the sequence $ x_n = \{ \alpha n^2\}$, where $\{ \}$ means fractional part: $\{ x \} = x -\lfloor x\rfloor$. Prove that for all $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ the set of ...
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  • 39
0 votes
1 answer
37 views

Proof of an inequality connected to Diophantine approximation

I'm recently doing some work that's vaguely connected with Dirichlet's approximation theorem. I came across this inequality that I haven't been able to prove. $$\forall\ a, b \in \mathbb{Z^+}, N\geq1,\...
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  • 5
3 votes
2 answers
67 views

Asymptote for $\frac{\sum _{j=1}^x \text{frac}\left(\frac{x}{j}\right)}{x}$?

I have been noodling around with the function $$f(x):=\frac{\sum _{j=1}^x \text{frac}\left(\frac{x}{j}\right)}{x}$$ where $x$ is a positive integer, and $\text{frac}(n)$ denotes the fractional part of ...
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