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Fractional Part Integral Open Problem

Find a closed form of

\[ \int_{0}^{1} \ldots \int_{0}^{1} \left \{ \dfrac{x_{1}}{x_{2}} \right \} \left \{\dfrac{x_{2}}{x_{3}}\right \} \ldots \left \{\dfrac{x_{n-1}}{x_{n}}\right \} \left \{\dfrac{x_{n}}{x_{1}}\right \} \ \mathrm{d}x_{1} \ \mathrm{d}x_{2} \ldots \mathrm{d}x_{n} \quad ; \quad n \geq 3 \]

Notation : \(\{ \cdot \}\) denotes fractional part function.


This is a part of the set Formidable Series and Integrals

Note by Ishan Singh
1 year, 7 months ago

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Damn. Its looks so difficult :D it is easier to solve a rubic's cube in 1 min rather solving this problem :p

Aman Rajput - 1 year, 7 months ago

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i was making good progress,but i got stuck at

\(\displaystyle\int _{ 0 }^{ 1 }{ \left\lfloor \frac { { x }_{ 1 } }{ { x }_{ 2 } } \right\rfloor \left\lfloor \frac { { x }_{ 3 } }{ { x }_{ 1 } } \right\rfloor d{ x }_{ 1 } } \)

:(

Hummus A - 1 year, 7 months ago

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okay i will try

Aman Rajput - 1 year, 6 months ago

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Case \(n=2\) is very simple.

Ishan Singh - 1 year, 7 months ago

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ya right :)

Aman Rajput - 1 year, 7 months ago

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