Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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24 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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1answer
129 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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0answers
65 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 ...
2
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1answer
56 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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2answers
76 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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0answers
31 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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1answer
36 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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2answers
78 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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0answers
40 views

When is $\sum_{n \in \mathbb{Z}} f(n) > \int_{\mathbb{R}} f(x) \, \mathrm{d}x$?

In relation to a research problem, I am facing the problem of showing that \begin{align} \sum_{n \in \mathbb{Z}} f(n) > \int_{\mathbb{R}} f(x) \, \mathrm{d}x \end{align} where $f$ is a specific non-...
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1answer
92 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 ...
6
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1answer
160 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 $a+b$, $a-c$,...
6
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2answers
236 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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3answers
64 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}...
6
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1answer
160 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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0answers
70 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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1answer
12 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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1answer
75 views

Suppose no children, we have 8 cookies. How many cookies does each child get?

Undeniably, I know that division by zero is undefined. But I don't understand how these stories teach this. JPBurke's answer doesn't my question. I simplified it. Your corrupt, profiteering ...
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0answers
15 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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1answer
90 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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89 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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1answer
115 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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2answers
245 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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1answer
84 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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34 views

Analyzing $f\circ g$ (involving fractional part and absolute value functions)

Given $$f:(-1,1)\to \mathbb{R } \ \ , \ \ g:(-1,1)\to \mathbb{(-1,1)}$$ $$f(x)=|2x-1|+|2x+1| \ \ , \ \ g(x)=\lbrace x \rbrace $$ Find the number of points of discontinuity and non-differentiability ...
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4answers
94 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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1answer
40 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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0answers
120 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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1answer
35 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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2answers
55 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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2answers
139 views

Evaluate the following integral: $\int_0^{\frac{\pi}{2}} \lbrace\tan x\rbrace\mathrm{d}x$ [duplicate]

$$I=\int_0^{\frac{\pi}{2}} \lbrace\tan x\rbrace\mathrm{d}x$$ So I have this interesting integral and I tried to evaluate it: Let $u=\tan x$ we'll have the following: \begin{align} \int_0^\infty \frac{\...
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0answers
49 views

Distributive Property of the Multiplication of Modulo Functions

According to this Wikipedia article, $ab \bmod n = [(a \bmod n)(b \bmod n)] \bmod n$. However $(x \bmod 1)^2 \neq (x^2 \bmod1)$ for $x \in \mathbb{R}$. So, is there some way to multiply modulo ...
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1answer
73 views

IGCSE Maths: Fractions Problem [closed]

I am currently trying to study for my IGCSE maths paper... I am stuck on this question ... can anybody help?? If 3/5 of people in a theatre buy a snack during ...
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1answer
93 views

Fractional part of a real number: questions

I was reading this question Evaluate the following integral $ \int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$ and I have seen that the user have used the mantissa of a real number or the ...
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3answers
289 views

Evaluate the following integral $ \int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$

$$\int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$$ Here $\lbrace\cdot\rbrace$ denotes the fractional part. I found this challenging integral, and I'm curious about the solution, so I decided ...
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2answers
74 views

Evaluating laplace transform of {x}

I came upon a question on Laplace transforms in my exams. $${\scr L}[\{x\}]$$ where $\{x\}=x - [x]$ denotes the fractional part of $x$. This is how I approached the problem. $$f(x)=\{x\}=\sum_{n=-\...
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1answer
19 views

Determining the integral part of the given term.

If $ m > 2$ and $t ∈ R$, find the integral part of $\left ( \frac{4|t|}{16+t^{2}} \right )^m$. I have approached the question by dividing the problem into cases when $-1 <t <1$, $t>1$ and $...
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1answer
122 views

Solving $f(f(f(x))) = 17$ when $f(x)=x\{x\}$

Can anybody help me in how to approach this problem. I expanded the fractional part of $x$ and tried to simplify but nothing is happening it is not coming in any format. For real number $x$ let $\...
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0answers
17 views

Finding integral of function involving fractional part of x [duplicate]

The integral given is: $$\int_{0}^{1}\big\lbrace\frac{1}{x}\big\rbrace \big\lbrace\frac{1}{1-x}\big\rbrace \big\lbrace1-\frac{1}{x}\big\rbrace dx$$ where $\big\lbrace x\big\rbrace$ represents the ...
2
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1answer
63 views

integral of fractional part $\int_0^1\left\{\frac 1x\right\}dx$ convergent?

$$I=\int_0^1\left\{\frac 1x\right\}dx=\int_1^\infty\frac{\{u\}}{u^2}du=\sum_{k=1}^\infty\int_0^1\frac{\{v+k\}}{(v+k)^2}dv=\sum_{k=1}^\infty\int_0^1\frac{v}{(v+k)^2}dv=\sum_{k=1}^\infty\ln\left(\frac{k+...
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3answers
283 views

Integral $\int_{0}^{1}\left\{\frac{1}{x}\right\}\left\{\frac{1}{1-x}\right\}\left\{1-\frac{1}{x}\right\} d x=?$

$\int_{0}^{1}\left\{\frac{1}{x}\right\}\left\{\frac{1}{1-x}\right\}\left\{1-\frac{1}{x}\right\} d x=?$ Where $\{\mathbf{x}\}$ is fractional part of $\mathbf{x}$ I reduced $\{1-1/x\}$ to $\{1/x\}$ but ...
5
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1answer
581 views

A sum involving fractional parts and prime numbers

In this paper a formula involving fractional parts, denoted by $\{\cdot\}$, is derived \begin{equation} \sum_{\;\;\;\;\;d\leq x \\ d \equiv b \mod a}\Big\{ \frac{x}{d}\Big\} = \frac{x}{a}(1-\gamma) + ...
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0answers
36 views

Concerning the fractional part function

I have that $f(x)$ and $g(x,y)$ are real polynomial with no constant terms. We have that $\forall \epsilon > 0$ the set $S_{\epsilon}$ = { $x \in\mathbb{Z} :$ Frac(f(x)) $ < \epsilon $ } is ...
6
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1answer
127 views

How to compute $\lim\limits_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$, where $\{x\}$ is the fractional part of $x$?

I've stumbled upon the following problem: $$\lim_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$$ where "{}" notates the fractional part. I've never studied this kind of problem, there exists any ...
0
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1answer
26 views

What will be the domain of the function $f(x)=\sqrt{2\{x\}^2-3\{x\}+1}$?

I need to find the domain of the function $$f(x)=\sqrt{2\{x\}^2-3\{x\}+1}$$ where $x \in [-1,1]$ and $\{.\}$ represents the fractional part of $x$ So here's what I tried: Clearly the part inside ...
9
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1answer
138 views

Solving $f(x+1)=xf(x)f\left(\frac1x\right)$

Recently, while scribbling around, I came up with the functional equation $f(x+1)=xf(x)f\left(\frac1x\right)$. It is somewhat similar to the functional equation for which the Gamma function is a ...
7
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1answer
88 views

On the asymptotics of $\sum_{k=1}^{n^2} \{\sqrt{k}\} $

On the asymptotics of $\sum_{k=1}^{n^2} \{\sqrt{k}\} $ This was inspired by a problem in Quora which asked to show that $s(n) =\sum_{k=1}^{n^2} \{\sqrt{k}\} \le \frac{n^2-1}{2} $. ($\{...\}$ means ...
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1answer
49 views

Fractional part integral involving Gamma function

Today there is a new integral with fractional part function… At least, this is the last integral and ufficially recognised by the author himself has the most difficult on his book. We have $$\int_0^1 \...
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1answer
47 views

Is there a way to test parity of fractional part (only period) of irredecible rational number without calculation?

I search in the web to get any way to test parity of fractional part of irredicible rational number by means to know if that fraction (period) even or odd but i didn't get , for example the fraction ...
0
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1answer
60 views

Another even worse double fractional part integral

These integrals are something terrible for me at the moment and the book where I pick them isn't very useful since the examples are quite repetitive in the chapter of Fractional Part Integrals… ...
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2answers
80 views

Another double fractional part integral

Today I'm trying to defeat another fractional part integral but it seems quite difficult… That is $$\int_0^1 \int_0^1 \left\{\frac{x+y}{x-y}\right\}\,\mathrm{d}x\,\mathrm{d}y$$ Probably, a way to ...

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