In [this question](https://mathoverflow.net/questions/216320/the-sum-of-a-series) the OP asks whether the sum
$$
f(q, \alpha) = \sum \_{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)\_k}
$$
is ever zero. An experiment with Mathematica indicates, to any reasonable precision that
$$f(2, 1) = \sum \_{k=1}^{\infty } \frac{2^k }{(2;2)\_{k-1}} = 0,$$
but I, for one, can't actually prove it. Is it true?



The sum of a series, continued
$$1/(2-1)\dots (2^{k-1}-1)+1/(2-1)\dots (2^k-1)=2^k/(2-1)\dots (2^k-1),$$
thus alternating sum of expression on the right is telescopical.


