Let r be the unique real number such that . 2.4413754618570030342085445640326490248139472302… or so.

There is an elegant relationship between and the lambert W function:

observe that if

Then

Which means that the infinite tetration of is , and

the infinite tetration or power tower is expressible in terms of the Lambert W function.

So we’ve got:

Now, if we move that to the other side, we get:

since latex r\log(r-1} = \log{r}$, so we

can rewrite the bottom like this:

Then:

Then

But since $latex , then

and finally

So, we’ve got

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