A nice series for b^^h , base sqrt(2), by diagonalization

[Gottfried]

I attempted a proof this way. Don't know, whether this is sufficient...

Using b=t^(1/t), u=log(t), log(b) = u/t , t being the fixpoint
Assume all the following lines as limit-expressions for @->0 (saving me typo-load)

Code:

´
   b^^(-2+@) - log_b(@)      = b^^inf   == fixpoint

   b^^(-2+@) - log(@)/log(b) = b^^inf
                             =  t

  log(b)* b^^(-2+@) - log(@) = u/t * t
                             = u

  log(b)* b^^(-2+@)          =  u +  log(@)

exponentiate

   b^(b^^(-2 + @)   = t * @
=   b^^(-1 + @)


exponentiate again, using base b

   b^(b^^(-1 + @))   = b^(t * @ )
                     = t^@
= b^^(0 + @)        = t^@

then this is

       lim @->0    b^^(0 + @) = 1
       lim @->0    t^@        = 1

saying

     lim { @->0 }  b^^(-2 + @ ) - log_b( @ ) = b^^inf

is correct in the limit.

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Is this -at least in principle - sufficient?

Gottfried