A386715 Decimal expansion of Integral_{x=0..1} {1/x}^2 * x^2 dx, where {} denotes fractional part.
0, 5, 1, 0, 0, 3, 0, 0, 9, 9, 9, 7, 3, 9, 3, 0, 9, 2, 7, 0, 9, 2, 8, 2, 2, 2, 3, 9, 4, 7, 5, 0, 8, 2, 7, 3, 3, 3, 8, 6, 8, 7, 9, 3, 5, 4, 8, 4, 2, 3, 4, 2, 2, 6, 8, 2, 4, 0, 5, 6, 7, 3, 8, 4, 2, 9, 3, 8, 4, 7, 7, 4, 6, 0, 3, 4, 9, 5, 3, 4, 5, 3, 2, 6, 6, 3, 8, 4, 0, 8, 5, 9, 0, 3, 0, 2, 0, 1, 2, 1, 8, 3, 2, 6, 5
Offset: 0
Examples
0.05100300999739309270928222394750827333868793548423...
References
- Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.21, pages 103 and 110.
Programs
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Mathematica
RealDigits[1 - (Zeta[2] + Zeta[3])/3, 10, 120, -1][[1]]
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PARI
1 - (zeta(2) + zeta(3))/3
Formula
Equals 1 - (zeta(2) + zeta(3))/3.
Equals 1 - A347213 / 3.
Equals Integral_{x=0..1} Integral_{y=0..1} {x/y}^2 * {y/x}^2 dx dy.
In general, for m >= 1, Integral_{x=0..1} {1/x}^m * x^m dx = Integral_{x=0..1} Integral_{y=0..1} {x/y}^m * {y/x}^m dx dy = 1 - Sum_{k=2..m+1} zeta(k)/(m+1).