Another proof of TPID 6

[tommy1729]

Conjecture

where such that

Discussion

To evaluate f at real numbers, an extension of tetration is required, but to evaluate f at positive integers, only real-valued exponentiation is needed. Thus the sequence given by the solutions of the equations

• 
  
• 
  
• 
  
• 
  

and so on... is the sequence under discussion. The conjecture is that the limit of this sequence is , also known as eta (). Numerical evidence indicates that this is true, as the solution for x in is approximately 1.44.

lim n-> oo x^^n = n conj : any real x = eta

since (eta+q) ^^ n grows faster than n for any positive q , we can use the squeeze theorem

lim q -> 0 eta =< x <= eta + q

hence x = eta

see also http://en.wikipedia.org/wiki/Squeeze_theorem

QED

regards

tommy1729