real slog developed at a fixed point

[Kouznetsov]

I do not understand why \( m\le n \)

Didnt you say that the series does not converge? Or not to the values of slog?

Not to the values of slog. The radius of convergence of the sub-series \( S(z) \) of terms with integer poers of \( z \) seems to be \( |L| \).
This subseries gives the approximation \( S_{+}(z) \) of \( \mathrm slog(z) \), which does not seem to be better than that with the sum of the first 5 terms (from m=0 to m=4).
I include below the similar figure for the approximation of slog with \( \frac{1}{L}\Big(\log(z-L) + \text{polynomial of 64th power of }(z-L) \Big) \).
   
Looking a the pic, I expect, the subseries has singularity at \( z=0 \). If we holomorphically exptend the subseries \( S \), I expect, \( \Re(S(0))=-1 \). But it is difficult to think that that \( \Im(S(0))=0 \).

Kneser klaims the "gleichmassig und absolut konvergeirt". Any subseries of the absloutely convergent series should converge too. Therefore, I guess, the radius of convergence of the Kneser's expansion is not larger than \( |L| \).

I tried to calculate the soefficients in terms with fractional powers of \( z-L \).
I failed to force Maple to the asymptotic analysis with fractional powers.
I should try Mathematica the same.