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Decimal expansion of a certain constant (see Comments lines for definition).
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%I #38 Sep 16 2024 14:26:16

%S 5,7,2,1,5,0,1,0,3,7,4,2,6,6,4,9,2,8,4,4,3,3,5,6,7,5,5,1,0,7,8,0,1,7,

%T 8,0,0,5,9,9,5,5,5,9,5,2,9,8,9,4,4,7,5,3,0,5,3,3,7,1,4,5,3,1,0,1,6,6,

%U 8,2,0,8,6,4,4,1,2,3,8,7,5,9,5,8,0,5,0,4,7,3,5,8,7,2,9,5,0,2,1,0,4

%N Decimal expansion of a certain constant (see Comments lines for definition).

%C Fanfolding the triangles defined by the spiral of Theodorus, each successive hypotenuse converges to a limit which divides the angle Pi/4 of the first triangle. The ratio of the limit angle to Pi/4 is 0.5721501037... = -(Sum_{n>=1} (-1)^n*arctan(1/sqrt(n))) / (Pi/4) The index n starts at 1 for the first triangle (Pi/4) and counts triangles.

%C I have developed an algorithm to compute the alternating sum, which converges quite slowly. I have computed a value to the precision limit of 64-bit floating point, which is about 14 decimal places. The execution time is about one second. It is unclear whether the algorithm readily adapts to arbitrary precision calculation software in a usable way. There are related sequences: the actual convergent angle in degrees, or radians, etc. as well as a whole family of similarly defined convergents. Other than general treatments of the spiral of Theodorus, no references are known.

%C Using the sumalt() function in PARI allows one to find as many terms as desired (see PARI program). - _Joerg Arndt_, Jan 05 2011

%F Equals -Sum_{n >= 1} (-1)^n*arctan(1/sqrt(n)) / (Pi/4).

%F Equals -argument(Product_{n >= 1} (sqrt(n)+((-1)^n)*i)/magnitude(sqrt(n)+((-1)^n)*i)) / (Pi/4).

%t digits = 101; -NSum[(-1)^n*ArcTan[1/Sqrt[n]], {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> digits+10]/(Pi/4) // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 15 2013 *)

%o (PARI) default(realprecision,155);

%o -sumalt(n=1,(-1)^n*(atan(1/sqrt(n))))/(Pi/4)

%o /* gives 0.57215010374266492844335675510780178005995559529894... */

%K cons,nonn

%O 1,1

%A Peter Hammer, Apr 14 2006

%E More terms from _Joerg Arndt_, Jan 05 2011