The series
 |
(1)
|
for 
an integer other than 0 and 
. 
and the related series
 |
(2)
|
which is a q-analog of the natural logarithm of 2, are irrational for 
a rational number other
than 0 or 
(Guy 1994). In fact, Amdeberhan and Zeilberger (1998) showed that the irrationality
measures of both 
and 
are 4.80, improving the value of 54.0 implied by Borwein (1991, 1992).
Amdeberhan and Zeilberger (1998) also show that the 
-harmonic series and q-analog
of 
can be written in the more quickly converging forms
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
where 
is a q-binomial coefficient and
is a 
-Pochhammer
symbol.
." J. Number Th. 37,
253-259, 1991.Borwein, P. B. "On the Irrationality of Certain
Series." Math. Proc. Cambridge Philos. Soc. 112, 141-146, 1992.Breusch,
R. "Solution to Problem 4518." Amer. Math. Monthly 61, 264-265,
1954.Erdős, P. "On Arithmetical Properties of Lambert Series."
J. Indian Math. Soc. 12, 63-66, 1948.Erdős, P. "On
the Irrationality of Certain Series: Problems and Results." In