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

[Gottfried]

Hmm,

this gives then also a strange formula for a limit.

It looks as if this would give

\( \lim_{ x->0} b\^\^^{\tiny{-2}} - \log_b(x) = b\^\^^{\tiny{\infty}} \)

Using b as base, b = t^(1/t) u=log(t), such that log(b) = u/t and t is a fixpoint the above series is formally

Code:

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  b^^h = t     - sum    t/u * coeff[k]/k  * (u^(h+2))^k
       = t     - sum    coeff[k]/k  *(u^(h+2))^k / log(b)

if h = -2 we get the zero'th powers of u (=1) at each coefficient and

Code:

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  b^^h = t     -  sum  coeff[k]/k* 1 / log(b)

and since the coefficients converge to 1 this is in principle in the limit a zeta(1)-series

Code:

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  b^^(-2) = t     - zeta(1) / log(b)      // limit h->-2

or the log(0) to base b

Code:

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  b^^(-2) = t +     log_b(0)

t is the fixpoint, so t = b^^inf and we have

Code:

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  b^^(-2) - log_b(0) = b^^inf   == fixpoint

or better expressed as limit

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

???
Gottfried