# Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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