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 Growth of superexponential tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 02/26/2013, 10:00 PM (This post was last modified: 02/26/2013, 10:26 PM by tommy1729.) 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 « Next Oldest | Next Newest »

 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

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