Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Growth of superexponential
#2
That is indeed trivial. In fact so trivial that I doubted if I would even respond.

Notice that for (real) b >> (z+1) exp(1/e) and (real) z > 0 :

b^^z > z.

Hence lim b^^z >= lim z.

If you are not convinced of b^^z > z Consider that

1) b^z > z for z and b sufficiently large.

2) b^^0 (=1) > 0

3) the derivative of b^^z > 1 whereas the derivative of z is 1.

By induction that is clear.

However I must add that I did assume a nice tetration here.

In other words I assumed for c>>b that c^^z >> b^^z and that c^^z and b^^z are continu.

It always matters what type of tetration you speak of.

However they share similar properties e.g. all infinitely differentiable real solutions to exp^[1/2](x) agree on their values infinitely often.

Notice it is nicer if a half iterate is strictly rising , otherwise when it is both rising and decending taking a half derivitive of that is troublesome.

On a piece of paper that might make more sense to you.

Perhaps usefull to note is that in your question your value of b does not depend on 1/z. That is important since b^^0 = 1 , so if z goes to 0 much faster than b goes to oo ...

regards

tommy1729
Reply


Messages In This Thread
Growth of superexponential - by Balarka Sen - 02/26/2013, 11:19 AM
RE: Growth of superexponential - by tommy1729 - 02/26/2013, 10:00 PM
RE: Growth of superexponential - by Balarka Sen - 02/27/2013, 02:19 PM
RE: Growth of superexponential - by sheldonison - 02/27/2013, 06:40 PM
RE: Growth of superexponential - by Balarka Sen - 02/27/2013, 07:24 PM
RE: Growth of superexponential - by tommy1729 - 03/01/2013, 12:11 AM
RE: Growth of superexponential - by tommy1729 - 03/06/2013, 11:51 PM
RE: Growth of superexponential - by tommy1729 - 03/06/2013, 11:55 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Between exp^[h] and elementary growth tommy1729 0 1,188 09/04/2017, 11:12 PM
Last Post: tommy1729
  Growth rate of the recurrence x(n+1) = x(n) + (arcsinh( x/2 ))^[1/2] ( x(n) )? tommy1729 0 1,841 04/29/2013, 11:29 PM
Last Post: tommy1729
  General question on function growth dyitto 2 3,969 03/08/2011, 04:41 PM
Last Post: dyitto
  Nowhere analytic superexponential convergence sheldonison 14 18,096 02/10/2011, 07:22 AM
Last Post: sheldonison
  The upper superexponential bo198214 18 21,271 09/18/2009, 04:01 PM
Last Post: Gottfried
  Question about speed of growth Ivars 4 6,115 05/30/2008, 06:12 AM
Last Post: Ivars
  Hilberdink: Uniqueness by order of growth? bo198214 2 3,741 05/30/2008, 12:29 AM
Last Post: andydude
  superexponential below -2 bo198214 10 9,820 05/27/2008, 01:09 PM
Last Post: GFR



Users browsing this thread: 2 Guest(s)