Write
. (Only later did I notice that it's actually
and not
, but I wanted to post this anyway.)
If we expand the infinite product
, a typical term is
, where
is a sum of powers of two.
The following is the
Basis Representation Theorem for base two.
Every positive integer
can be represented
uniquely as the sum of powers of two. Example:
.
Since the exponents of the terms in the infinite series
consist of
all possible combinations of powers of two, it follows from the theorem that the exponents range over
all the positive integers.
Therefore,
.
We have
, i.e.
.
Here,
.
So, a third one?