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A088538
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Decimal expansion of 4/Pi.
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43
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1, 2, 7, 3, 2, 3, 9, 5, 4, 4, 7, 3, 5, 1, 6, 2, 6, 8, 6, 1, 5, 1, 0, 7, 0, 1, 0, 6, 9, 8, 0, 1, 1, 4, 8, 9, 6, 2, 7, 5, 6, 7, 7, 1, 6, 5, 9, 2, 3, 6, 5, 1, 5, 8, 9, 9, 8, 1, 3, 3, 8, 7, 5, 2, 4, 7, 1, 1, 7, 4, 3, 8, 1, 0, 7, 3, 8, 1, 2, 2, 8, 0, 7, 2, 0, 9, 1, 0, 4, 2, 2, 1, 3, 0, 0, 2, 4, 6, 8, 7, 6, 4, 8, 5, 8
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OFFSET
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1,2
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COMMENTS
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Average length of chord formed from two randomly chosen points on the circumference of a unit circle (see Weisstein/MathWorld link). - Rick L. Shepherd, Jun 19 2006
Suppose u(0) = 1 + i where i^2 = -1 and u(n+1) = (1/2)*(u(n) + |u(n)|). Conjecture: limit(Real(u(n)), n = +infinity) = 4/Pi. - Yalcin Aktar, Jul 18 2007
Ratio of the arc length of the cycloid for one period to the circumference of the corresponding circle rolling on a line. Thus, for any integral number n of revolutions of a circle of radius r, a point on the circle travels 4/Pi*2Pi*r*n = 8rn (while the center of the circle moves only 2Pi*rn). This ratio varies for partial revolutions and depends upon the initial position of the point with points nearest the line moving the slowest (see Dudeney, who explains how the tops of bicycle wheels move faster than the parts nearest the ground). - Rick L. Shepherd, May 05 2014
Average distance traveled in two steps of length 1 for a random walk in the plane starting at the origin. - Jean-François Alcover, Aug 04 2014
This is also the value of a special case (n=1) of an n-family of series considered by Hardy (see A278145): 1 + (1/2)*(1/2)^2 + (1/3)*(1*3/(2*4))^2 + (1/4)*((1*3*5) / (2*4*6))^2 + ... = Sum_{k>=0} (1/(k+1))*((2*k-1)!!/(2*k)!!)^2. - Wolfdieter Lang, Nov 14 2016
Minimum ratio of the area of a rectangle to one of its inscribed ellipses, or only existing ratio if the rectangle is a square and then the only inscribed ellipse is a circle. This ellipse has its semiaxes parallel to the sides of the rectangle. If a rectangle has sides of length 2a and 2b, its area is 4*a*b, while the ellipse inscribed has a and b as semiaxes, therefore its area is a*b*Pi. Thus the ratio is (4*a*b)/(a*b*Pi) = 4/Pi. - Giovanni Zedda, Jun 20 2019
The diameter of the conventional spherical earth is (4/Pi)*10000 km = 12732.395... km. - Jean-François Alcover, Oct 30 2021
sign(sin(x)) = (4/Pi) * Sum_{n>=0} sin((2*n+1)*x)/(2*n+1), for all x in R;
sign(cos(x)) = (4/Pi) * Sum_{n>=0} (-1)^n*cos((2*n+1)*x)/(2*n+1), for all x in R. (End)
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REFERENCES
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H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp. 99, 300-301, #294.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 86
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 105, eq. (7.5.1) for n=1.
L. B. W. Jolley, Summation of Series, Dover (1961).
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LINKS
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Eric Weisstein's World of Mathematics, Cycloid.
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FORMULA
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4/Pi = Product_(1-(-1)^((p-1)/2)/p) where p runs through the odd primes.
Arcsin x = (4/Pi) Sum_{n = 1, 3, 5, 7, ...} T_n(x)/n^2 (Chebyshev series of arcsin; App C of math.CA/0403344). - R. J. Mathar, Jun 26 2006
Equals 1 + Sum_{n >= 1} ((2n-3)!!/(2n)!!)^2. [Jolley eq 274]. - R. J. Mathar, Nov 03 2011
Equals 1 + Sum_{n>=0} (Catalan(n)/2^(2*n+1))^2 , with Catalan(n) = A000108(n). This is the rewritten Jolley (274) series. See the above R. J. Mathar entry with (-1)!! := 1. - Ralf Steiner, Sep 18 2018
4/Pi = 1 + (1/4)*hypergeometric([1, 1/2, 1/2], [2, 2], 1) = hypergeometric([-1/2, -1/2], [1], 1). From the g.f. of Catalan^2 given in A001246. - Wolfdieter Lang, Sep 18 2018
Equals Product_{k>=1} (1 + 1/(4*k*(k+1))). - Amiram Eldar, Aug 05 2020
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EXAMPLE
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4/Pi = 1.2732395.... = 1/0.78539816...
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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