Find a closed form of ∫01…∫01{x1x2}{x2x3}…{xn−1xn}{xnx1} dx1 dx2…dxn;n≥3 \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 ∫01…∫01{x2x1}{x3x2}…{xnxn−1}{x1xn} dx1 dx2…dxn;n≥3
Find a closed form of
∫01…∫01{x1x2}{x2x3}…{xn−1xn}{xnx1} dx1 dx2…dxn;n≥3 \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 ∫01…∫01{x2x1}{x3x2}…{xnxn−1}{x1xn} dx1 dx2…dxn;n≥3
Notation : {⋅}\{ \cdot \}{⋅} denotes fractional part function.
This is a part of the set Formidable Series and Integrals
Note by Ishan Singh 5 years, 2 months ago
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2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Damn. Its looks so difficult :D it is easier to solve a rubic's cube in 1 min rather solving this problem :p
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Case n=2n=2n=2 is very simple.
ya right :)
i was making good progress,but i got stuck at
∫01⌊x1x2⌋⌊x3x1⌋dx1\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 } } ∫01⌊x2x1⌋⌊x1x3⌋dx1
:(
okay i will try
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Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
*italics*
or_italics_
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or__bold__
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to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
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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
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Case n=2 is very simple.
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ya right :)
i was making good progress,but i got stuck at
∫01⌊x2x1⌋⌊x1x3⌋dx1
:(
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okay i will try