martin Wrote:By the way, I achieved slightly better results when I changed that n=0.345627(2-x) into n=0.691-0,3450832*x. Gosh, I think this is getting a bit out of hand here...

Well this looks very much like ln(2) - ln(2)/2*x ....

May be it is worth trying ( just out of pure interpolation) without really following what has been done in this thread) something like

ln(2)-ln(2)/2*x+(ln(2)/3!)^x^2 - (ln(2)/4!)*x^3 +

or similar (may be not factorial, but just 1/2, 1/3, 1/4, etc) alternating series with ln(2) in coefficient.

The inversed Euler limit it very interesting because it involves ROOTs and roots can have many values- these in the end should correspond to the infinitely many values of complex logarithm, the thing I tried to understand here:

Cardinality of infinite number of values of complex logarithm via ROOTS of I

As You take the limit of inversed formula, the number of roots with n-> infinity grows (since roots are n-valued) as does the possible combinations of the values. However , there are infinitely many ways n can reach infinity via subsets of N ( like 2^n, n^2, primes, n, etc etc) each way generating a different tree structure, and each of them corresponding to convergent or divergent series over 1/n , 1/(2^n), 1/(n^2) so root series are such may converge or diverge ( if they are placed in exponents via product of roots of I in complex logarithm, as I did in the tread I mentioned above, these sums make certain sense, but they do not hold all information- neither about TREE structures underlaying them). If we turn this around, it may be that this underlaying Tree structure is the cause and reason for divergence/convergence if certain sums.

In the end infinite values of complex logarithm obtained via roots should EXACTLY match the infinite number of values complex Ln(x) can have, but that means the infinite number of values of complex logarithm has rather clear internal relations via TREE structure related to the combinations of multivalued roots- I do not know if that is made explicit somewhere.

My opinion is that complex logarithm ( as imaginary unit itself) is far from being understood.

Ivars