# Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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### 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 ...
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### 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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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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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