True or False Logarithm

[Henryk Trappmann]

Let |b|<1 and consider the following polynomial sequence f_n(x):

f_n(x) = -sum from k=1 to n: binomial(n over k)*(-1)^k * (1-x^k)/(1-b^k)

I can show by elementary transformations that
lim_{n -> oo} f_n(b^m) = m for every integer m>=0 and
lim_{n -> oo} f_n(0) = -oo

Does this function sequence converge also for other points |x|<1 than
b^m and is the limit log_b(x)?

This function sequence looks too elementary, it must have been
considered already in the history of arithmetics (Euler?). Does it
remind anybody of something?