iterating x + ln(x) starting from 2

[tommy1729]

consider the sequence starting with x=2 then 2+ln(2) and taking x+ln(x) at every step.

Now a brute estimate of the sequence would be x + n ln(x) where x=2 and n is the n th iterate.

However that is not accurate.

First the sequence grows approximately like x + n ln(n) ln(x) on average.

Later like x + n (ln(n) + ln(ln(n))) ln(x).

And it continues like x + n (ln(n) + ln^[2](n) + ln^[3](n)) ln(x).

The pattern is clear. However it seems to be most regular for values of x near x = 2. Fascinating.

I guess there is a simple reason for this behaviour , right ?

It seems like selfreference almost.

guess this is in many dynamical systems books but maybe not.

regards

tommy1729

[tommy1729]

Apparantly my friend mick posted a more formal (and slightly stronger) version of this question to MSE.

http://math.stackexchange.com/questions/...with-f-0-2

However it is not very popular and it even got closed.

On the other hand it also got votes to be reopened and I think his table is correct.

Perhaps my OP here was to vague and some of you might benefit from the post by mick.

To see the link notice that ln(x) + ln^[2](x)+... is close to arcsinh(x/2) + archsinh... because ln(x) is close to arcsinh(x/2).

***

I must say im not surprised that it got closed, but not for the right reasons imho.
The attitude towards tetration and some parts of dynamical systems is unacceptable and prehistoric imho.
This imho fact is one of the reasons i came to the tetration forum. But i guess i should stop talking now because im getting emotional.
However I guess some ppl including mick, lwalke3, john97, bill dubuque, timothy golden, quasi, galathaea, jasper74 and many more ( including some members here and ppl who insisted me to (*) ) understand why I did not become a member of /or removed my membership of/ some chat/Q&A/math sites.

I know im not the first to complain about this type of behaviour towards tetration and those posting about it , but anno 2013 the situation has not improved much although dynamical systems has gained popularity.

(* clear from context)

regards

tommy1729

[tommy1729]

By deeper understanding of this ( the question can be done by standard techniques ) , I came (with mick) to a more refined and relevant ( to tetration ) conjecture. That conjecture has been posted by me here on the tetration forum.

Link :

http://math.eretrandre.org/tetrationforu...hp?tid=790

Regards

tommy1729