Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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1answer
29 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=-\...
0
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1answer
17 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 $...
0
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0answers
23 views

Period of composite function involving fractional part.

Let $\displaystyle f(x)=\sum_{r=0}^{n-1} \left\{\frac{x+r}{n}\right\}$, then fundamental period of $f(x)$ is? (where $\{.\}$ represents fractional part function). Here's the image of the question. My ...
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0answers
14 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
56 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+...
1
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3answers
220 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 ...
4
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1answer
52 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
31 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 ...
4
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1answer
67 views

How to compute $\lim_{n\rightarrow \infty} \{(2+\sqrt{3})^{n}\}$, 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
20 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 ...
7
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0answers
88 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
70 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 ...
1
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1answer
29 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
30 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
43 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… ...
0
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2answers
58 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 ...
2
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1answer
86 views

Evaluating $\int_0^n \{x^2\}\,\text{d}x$

I have problems with the series that appear evaluating this integral $$\int_0^n \{x^2\}\,\text{d}x$$ where $\{\cdot\}$ is the fractional part function. Now, I'm pretty sure that $$I=\sum_{k=0}^{n^2-1}\...
2
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3answers
70 views

Show that $|\{x\}^2-\{x\}+1/6|\leq \frac{1}{6}$

I am trying to show the following inequality holds for $x>0$: $$|{x}^2-{x}+1/6|\leq \frac{1}{6}.$. I was able to show that $\{x\}^2-\{x\}+1/6 \leq 1/6$ because $\{x\}^2-\{x\}<0.$ However I am ...
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2answers
29 views

Considering this constraint, determine the real function $f$

The problem Let $f: \mathbb{R} \to \mathbb{R}$ Determine $f(x)$ knowing that $ 3f(x) + 2 = 2f(\left \lfloor{x}\right \rfloor) + 2f(\{x\}) + 5x $, where $ \left \lfloor{x}\right \rfloor $ is the ...
3
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2answers
52 views

Upper bound on summation involving fractional part

Let $x\in[1,+\infty)\subset\mathbb{R}$. I would like to show that $$\sum_{d=1}^{\lfloor x\rfloor}\left(\frac{x}{d}-\left\lfloor\frac{x}{d}\right\rfloor\right)\leq x-1,$$ where $\lfloor\cdot\rfloor$ is ...
-1
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1answer
70 views

Find all real numbers $a$ s.t. $\sum_{k=0}^{2019}\left\lfloor x+\frac{k}{a}\right\rfloor=\lfloor ax\rfloor$ for every $x\in\Bbb R$. [closed]

Find all real numbers $a$ wiith $a\neq 0$ and $$\left \lfloor x \right \rfloor+\left \lfloor x+\frac{1}{a} \right \rfloor+\left \lfloor x+\frac{2}{a} \right \rfloor+\ldots+\left \lfloor x+\frac{2019}...
2
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0answers
54 views

On the convergence rate of $\sum \frac{\ln(n)}{n}\{x^n+x^{-n}\}$

Let $$S:=\left\{x\in \mathbb{R}:\sum \frac{\ln(n)}{n} \{x^n+x^{-n} \}<+\infty\right\}\\ S':=S\cap(1,+\infty)$$ In this post it was proved that $\mu(S)=0$, thanks to the equidistribution of $\{x^n\}...
7
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2answers
136 views

What is the behaviour of the first $n$ digits of ${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}$ as $n\to\infty$

For a natural number $n$, let $f(n)$ denote the first $n$ digits of the decimal expansion of $${\underbrace{99\dots99}_{n\text{ nines}}}^{\overbrace{99\dots99}^{n\text{ nines}}}=(10^n-1)^{10^n-1}.$$ ...
-3
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1answer
44 views

Try to prove the inequality [closed]

My problem: Show that the inequality is valid for all sufficiently large $n$ $$ \left\lbrace\frac{1}{{1}^{1/3}}+\frac{1}{{3}^{1/3}}+\cdots+ \frac{1}{{(2n+1)}^{1/3}}\right\rbrace>\frac12 $$ ...
2
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1answer
43 views

What is the standard notation for the set of rationals with finite fractional part?

As the title says. What is the standard notation for the set of rationals with finite fractional part, when written in base $n$ with a radix point? I expected $ℚ_n$, but that's taken for n-adic ...
1
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0answers
94 views

Find the asymptotic expansion as $N \rightarrow \infty$ of $\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \left\{{\sqrt{{N}^{2}+{k}^{2}}}\right\}$

This expression comes from the asymptotic expansion of $$\sum_{k=1}^{\left\lfloor{N/2}\right\rfloor} \sum_{i=1}^{\left\lfloor{\sqrt{{N}^{2}+{k}^{2}}}\right\rfloor - k} \tau \left({i \left({2\, k + i}...
9
votes
1answer
87 views

Limit of alternating sum of fractional parts

Find $\newcommand{\pars}[1]{\left\{ \frac{n}{#1} \right\}}$ $$\lim_{n\to\infty}\dfrac{1}{n} \left( \pars{1} - \pars{2} + ... + (-1)^{n+1} \pars{n} \right),$$ where $\left\{ x \right\} $ denotes the ...
2
votes
2answers
32 views

Limit of the decimal part of $\sqrt{4n^2+38n}$

Let $x_n=4n^2+38n$. Find the limit of the decimal part of $\sqrt{x_n}$ as $n$ goes to infinity. I have tried completing the square for $x_n$, which yields $\sqrt{x_n}<2n+\frac{19}{2}$, but that's ...
7
votes
1answer
197 views

On the convergence of $\sum\frac{\log(n)}{n}\{x^n+x^{-n}\}$

As stated in the title, I'm trying to determine the values of $x\in \mathbb{R}$ for which $$\sum_{n=1}^\infty \frac{\log(n)}{n}\{x^n+x^{-n}\}<+\infty$$ where $\{x\}$ is the fractional part (any ...
1
vote
2answers
85 views

Find $\lim\limits_{n \to \infty} n \int\limits_0^\pi \{ -x\}^n dx$.

I have to find the limit: $$\lim_{n \to \infty} n \int_0^\pi \{ -x \}^n dx$$ Where $\{ x \}$ is the fractional part of the real number $x$. I know that: $$\{ x \} = x - \lfloor x \rfloor$$ where $...
2
votes
1answer
117 views

Closed expression for sum $\sum_{k = 1}^{\infty} \frac{\left\lfloor \sqrt{k} \right \rfloor}{k^2}$

Generalizing a recent post Closed expression for sum $\sum_{k=1}^{\infty} (-1)^{k+1}\frac{\left\lfloor \sqrt{k}\right\rfloor}{k}$ where convergence was assured by alternating the sign here's a similar ...
1
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0answers
32 views

Continous, Non-Fourier, Trigonometric Approximation of the Fractional Part Function [duplicate]

I would like to know a single (not piecewise) continuous approximation of $x \bmod 1$ which gets sharper the more you increase a constant $c$. I do not want a series like Fourier, but I do want ...
0
votes
0answers
27 views

How to find if this function is periodic

Let $f:\mathbb R\to \mathbb R, f(x) =\{\frac{x+1}{3}\}$, where { } is the fractional part. So pretty much I have to find if this function is periodic. My first thought was this: We have ax + b so f ...
0
votes
0answers
45 views

Rewriting the Floor function in terms of the Heaviside Step Function and/or Sign Function

Is there a way to rewrite $\lfloor x \rfloor$ in terms of the Heaviside Step Function and/or Sign Function, or approximating it using them?
3
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2answers
71 views

Bound for sum of fractional parts of $\sqrt{k}\, \alpha$

Let us consider the sum $$C_\alpha(n)=\sum_{1\leq k\leq n} \left(\{ k\alpha\}-\frac{1}{2}\right)$$ where $\alpha$ is a positive irrational real number and $\{ x \}$ denotes the fractional part of $x$...
3
votes
1answer
102 views

Integration by parts for fractional Ornstein-Uhlenbeck process

So I have encountered a problem in a paper called Volatility is rough by Jim Gatheral et al. A stationary fractional Ornstein–Uhlenbeck process ($X_t$) is defined as the stationary solution of the ...
0
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1answer
42 views

Find number of x such that $\{x\}+\{x^2\}=1,x\in(0,20). $Where $\{\}$ denotes fractional part. [closed]

Find number of x such that $\{x\}+\{x^2\}=1,x\in(0,20).$Where $\{\}$ denotes fractional part. I did it by lot of case work searching for some simpler solution
4
votes
4answers
149 views

Evaluate the limit $\lim\limits_{n\to0}\frac{(x)+(2x)+\cdots (nx)}{n^2}$

Find the limit of $\lim_{n\rightarrow ~0}\frac{(x)+(2x)+\cdots (nx)}{n^2}$, where, $(x)=x-[x]$ and $[x] $ is the greatest integer function(the fractional part function). I feel, as $n \rightarrow 0$ ...
2
votes
1answer
81 views

Prove that $\left|\left\{\frac{n}{1}\right\} - \left\{\frac{n}{2}\right\} - \cdots - (-1)^n\left\{\frac{n}{n}\right\}\right| \le \sqrt{2n}$.

For all positive integers $n$, prove that $$\large \left|\left\{\frac{n}{1}\right\} - \left\{\frac{n}{2}\right\} + \left\{\frac{n}{3}\right\} - \cdots - (-1)^n\left\{\frac{n}{n}\right\}\right| \le \...
1
vote
0answers
87 views

Is fractional part of $e^n$ dense in (0,1)?

Can fractional part of $e^n$ where $n\in\mathbb{Z}_+$ dense in $(0,1)$? Generally, for which kind of $\alpha > 0$, $\{\alpha^n\}$ dense in $(0,1)$?
1
vote
3answers
48 views

If $\{x\}:=x-\lfloor x\rfloor$, then are these the same? $\{l+m\}$, $\left\{l+\{m\}\right\}$, $\left\{\left\{l\right\}+\{m\}\right\}$

I am currently working on some signal processing project and come across this particular problem: Define $\{x\} := x - \lfloor x\rfloor$ and consider $$y_1 = \{l+m\}$$ $$y_2=\left\{l+\{m\}\...
3
votes
1answer
39 views

If matrix $A=[a_{ij}]_{4 \times 4}$ such that…

If matrix $A=[a_{ij}]_{4 \times 4}$ such that $a_{ij}= \begin{cases} 2&\text{if}\, i=j\\ 0&\text{if}\, i \not= j\\ \end{cases}$ , then $\{\frac{det(adj(adjA))}{7}\}$ is ($\{x\}$ ...
-2
votes
1answer
137 views

Let $F_1$ and $F_2$ be the fractional parts of $(44-\sqrt 2017)^{2017}$ and $(44+ \sqrt 2017)^{2017}$ respectively. Then $F_1+F_2$ lies between? [closed]

The fractional part of a real number x is $x –[x]$, where $[x]$ is the greatest integer less than or equal to $x.$ Let $F_1$ and $F_2$ be the fractional parts of $(44-\sqrt 2017)^{2017}$ and $(44+ \...
1
vote
3answers
60 views

Prove $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$

Prove the following for all real $x$ i. $⌊2x⌋+⌊2y⌋≥⌊x⌋+⌊y⌋+⌊x+y⌋$ ii. $⌊x⌋-2⌊x/2⌋$ is equal to either $0$ or $1$ For ($ii$.) I attempted to split it into cases of whether the fraction part {$x$} is ...
0
votes
1answer
36 views

fractional imaginary calculation

Assume $j=\sqrt{-1}$, to calculate $(-3.1416j)^{0.5}$ in Matlab, I type (j*-3.1416)^0.5 and it gives me the result: ...
3
votes
2answers
54 views

Finding the limit of this recurrence relation

Let's say we have the equation $$a_{n+1} = Frac\left(\frac{1}{\ln(1+a_n)}\right)$$ with $a_0\ne 0$ and that it is not negative. I've tried finding the limit of $a_n$ as $n \rightarrow \infty$, but I ...
1
vote
2answers
89 views

Find the total number of solution of the equation

Find number of solution of :- $$[x]^2 = x + 2\{x\}$$ where $[.]$ represents greatest integer function and $\{.\}$ represents fractional part.
3
votes
1answer
72 views

Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
1
vote
1answer
26 views

proving floor, and rounding floats in some programming languages

Here is a simple property which is very useful while programming : In most programming languages, ...
2
votes
2answers
91 views

Counting the number of integers with their least prime factor greater than $x$ between $ax$ and $ax+x$

Let: $x \ge 2, a \ge 1$ be integers. $x\#$ be the primorial for $x$ $\mu(i)$ be the möbius function. $\text{lpf}(x)$ be the least prime factor of $x$. $p_k$ be the $k$th prime which is the highest ...

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