From habsiege@math.u-bordeaux.fr Tue Dec  8 12:44:06 1998

Dear Doron,

I have been reading your paper with Amdeberhan and it
reminded me a reference you might be interested in :
Daniel Duverney, A propos de la serie $\sum_{n=1}^{\infty} \frac{x^n}{q^n-1}$,
Journal de theorie des nombres de Bordeaux 8 (1996), 173-181.
If your library does not have this journal, let me know, I can send you
a xerox copy or the author adress.
He has general irrationality measures for this series and specific ones
for the q-harmonic series ; for instance, when q is an integer, 
$\sum_{n=1}^{\infty} \frac{1}{q^n-1}$ has an irrationality measure 4.31012...