I was tinkering with the question whether the logarithm $\log(x)$ can be expressed by some more useful series than by the Mercator series (in terms of (1+x)) for a certain question.
One idea was to try, what a representation as "iteration-series" would look like, where I write "iteration series" for a series composed by the iterates of a function (not as a series of powers of its argument $x$).
Here I write the iterate of a function $f(x)$ as
$ \qquad \qquad f^{°2}(x)\overset{def}=f(f(x)) \\ \qquad \qquad f^{°k+1}(x) = f^{°k}(f(x)) \\ \qquad \qquad f^{°1}(x)=f(x), f^{°0}(x)=x $
Then I try the following naive ansatz:
$$ \begin{array} {ll} \log(x) &= f(x) + f^{°2}(x) + f^{°3}(x) + ... \\ \log(f(x)) &= f^{°2}(x) + f^{°3}(x) + ... \\ f(x) &= \log(x) - \log(f(x)) \\ e^{f(x)} &= x / f(x) \\ x &= f(x) \cdot e^{f(x)} \end{array}$$ This has the form of the Lambert-W-function, so a conclusion should be $$ W(x)\overset{def}: W(x) \cdot e^{W(x)} = x \to f(x) = W(x) $$ However, the result is not correct; simply insert $x=0$ to see the contradiction; also the lhs in the first equation is likely divergent, and approximations using $x=2$ or similar show wrong results.
Q: So how could I repair this ansatz? Is there a correction, say in modification of the argument $x$ or of the iteration-series itself?
[update] Using an alternating iteration-series and a recentering of the logarithm to write $\log(1+x)$ I get a quickly converging series by $$ \begin{array} {ll} \log(1+x) &= x - f(x) + f^{°2}(x) - f^{°3}(x) + ... \\ \log(1+f(x)) &= f(x) - f^{°2}(x) + f^{°3}(x) - ... \\ x &= \log(1+x) + \log(1+f(x)) \\ e^x &= (1+x) \cdot (1+f(x)) \\ f(x) &= {e^x\over 1+x} -1 \end{array}$$
and for small values $x=2$ and $x=3$ which are of interest for me, the iteration-series made by this converges to many digits with a handful of terms: below are the first few partial sums for $\log(2)$:
k partial sums signed k'th iterate of f(1)
[0, 1, 1 ]
[1, 0.6408590857704774, -0.3591409142295226]
[2, 0.6945383440651704, 0.05367925829469305]
[3, 0.6931462124431120, -0.001392131622058478]
[4, 0.6931471805604139, 0.0000009681173019793587]
[5, 0.6931471805599453, -4.686252527398870 E-13]
However, I am searching for a non-alternating series...