open problems survey

[tommy1729]

We have the sequence of "Eulers" {2.718281828,3.0885322718...} and the sequence of "Etas" {1.44466786,1.6353244967...}.

"Eulers" and "Etas" can be defined as the x-coordinate and y-coordinate of the maximum of the nth order self root.

Conjecture:

The limit of the sequence of "Eulers" is 4.
The limit of the sequence of "Etas" is 2.

Some discussion can be found here

If you can find a better name for these sequences feel free to use it.

let the "Eulers" be eul(n) and the "Etas" be et(n).

now i conjecture :

1) et(n)^2 < eul(n-1)

2) lim n-> oo (et(n)^2 - eul(n-1)) / (et(n-1)^2 - eul(n-2)) = 1

regards

tommy1729