A224842 Decimal expansion of log((1+e)/2).

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

Offset

0,1

Comments

Positive matrices do not always behave like positive numbers: for instance 0 <= A <= B does not imply e^A <= e^B [e^A stands here for matrix exponential of A, and 0 <= A <= B means that A and B-A are positive matrices, a positive matrix being a matrix which has nonnegative entries]. The counterexample given by Paul Halmos is A = e^{{1/2, 1/2}, {1/2, 1/2}} = {{(1+e)/2, (e-1)/2}, {(e-1)/2, (1+e)/2}} and B = {{e^x, 0}, {0, e^y}}, with x < log((1+e)/2).

References

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 9Q (Exponential inequality) p. 273.

Links

Table of n, a(n) for n=0..99.

Formula

Equals Integral_{x=0..1} 1/(exp(-x) + 1) dx. - Amiram Eldar, Aug 16 2020

Example

0.620114506958277524631763373509679073839779951310093120598383505343801279539...

Mathematica

RealDigits[Log[(1 + E)/2], 10, 100][[1]]

Crossrefs

Sequence in context: A123147 A119831 A369016 * A318138 A130143 A195486

Adjacent sequences: A224839 A224840 A224841 * A224843 A224844 A224845

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 22 2013

Status

approved