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A322712
Decimal expansion of Sum_{k = -infinity .. infinity} exp(-k^2/4) - Integral_{x = -infinity .. infinity} exp(-x^2/4) dx.
0
5, 0, 7, 4, 2, 9, 8, 4, 5, 8, 4, 5, 7, 9, 5, 6, 9, 8, 0, 8, 8, 0, 5, 7, 0, 9, 4, 8, 3, 4, 2, 0, 1, 2, 0, 4, 5, 5, 1, 7, 9, 0, 8, 0, 3, 4, 5, 1, 5, 9, 0, 0, 4, 1, 2, 9, 9, 9, 9, 4, 0, 6, 0, 9, 2, 0, 9, 3, 2, 2, 5, 5, 3, 1, 1, 0, 8, 1, 0, 6, 4, 4, 5, 3, 7, 0, 5
OFFSET
-16,1
COMMENTS
This constant is the small difference between the sum and integral of the same function. The integral is 3.54490770181103205... (10 * sqrt(Pi)/5, see A019707) and the sum agrees up to 15 decimal digits, 3.54490770181103210... This approximation is similar to exact identities of sum and integral of the same function known as "Sophomore's dream" (A073009, A083648).
LINKS
Nick Lord, Solution to problem 81.F, The Mathematical Gazette, Vol. 82, No. 493 (1998), pp. 130-131.
T. J. Osier, Get billions and billions of correct digits of pi from a wrong formula, Mathematics and Computer Education, Vol. 33 (1999), pp. 40-45.
FORMULA
Equals 2 * sqrt(4*Pi) * Sum_{k >= 1} exp(-4 * Pi^2 * k^2) ~ 2 * sqrt(4*Pi) * exp(-4*Pi^2).
EXAMPLE
5.0742984584579569808805709483420120455179080345159... * 10^(-17).
MATHEMATICA
s = Sum[Exp[-n^2/4], {n, -Infinity, Infinity}] - Sqrt[4 * Pi]; RealDigits[s, 10, 100][[1]]
PROG
(PARI) default(realprecision, 100); 2*sqrt(4*Pi)*suminf(k=1, exp(-4*Pi^2*k^2)) \\ Michel Marcus, Dec 25 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Dec 24 2018
STATUS
approved