real slog developed at a fixed point

[bo198214]

I do not understand why \( m\le n \). The sum with first 5 terms with \( n=0 \) gives the approximation of slog with function \( f \)
\( f=f(z)=\frac{1}{c}\log(z-c)+\sum_{m=0}^4 c_{m,0} (z-c)^{m} \)
shown in the figure below.

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

Beautiful picture btw.