A322712 Decimal expansion of Sum_{k = -infinity .. infinity} exp(-k^2/4) - Integral_{x = -infinity .. infinity} exp(-x^2/4) dx.

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

Table of n, a(n) for n=-16..70.

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

Cf. A019707, A073009, A083648.

Sequence in context: A168195 A333507 A345327 * A244345 A021873 A020801

Adjacent sequences: A322709 A322710 A322711 * A322713 A322714 A322715

Keyword

nonn,cons

Author

Amiram Eldar, Dec 24 2018

Status

approved