# Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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