A135002 Decimal expansion of 2/e.

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

Offset: 0

Author

Omar E. Pol, Nov 15 2007

Keywords

• nonn
• cons

Comments

From Johannes W. Meijer, Jun 27 2016: (Start)

This constant is related to the values of zeta(2*n-1) of the Riemann zeta function and the Euler Mascheroni constant gamma. If we define Z(n) = (1/n) * (sum(zeta(2*n-2*k-1) * Z(k), k=0..n-2) + gamma * Z(n-1)), with Z(0) = 1, then limit(Z(n), n -> infinity) = 2/exp(1).

Similar formulas appear in A090998 and A112302.

The structure of the n! * Z(n) formulas leads to the multinomial coefficients A036039. (End).

Examples

			0.735758882342... = 2*A068985.
		

Links

• Index entries for transcendental numbers

Crossrefs

Cf. A036039, A068985, A112302, A090998.

Programs

Maple

evalf(2/exp(1)) ; # R. J. Mathar, Jul 14 2013

Mathematica

RealDigits[2/E,10,120][[1]] (* Harvey P. Dale, Dec 25 2013 *)

PARI

2*exp(-1) \\ Charles R Greathouse IV, Apr 16 2015

Formula

Integral of log x from x = 1/e to e. - Charles R Greathouse IV, Apr 16 2015

Equals lim_{k->0} 2*(1 - k)^(1/k). - Ilya Gutkovskiy, Jun 27 2016

Equals Sum_{i>=0} ((-1)^i)(1-i)/i!. - Maciej Kaniewski, Sep 10 2017

Equals Sum_{i>=0} ((-1)^i)(i^2+2)/i!. - Maciej Kaniewski, Sep 12 2017

From Peter Bala, Mar 21 2022: (Start)

2/e = Integral_{x = 1..oo} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx;

2/e = - Integral_{x = 0..1} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx, both valid for n >= 2. (End)