Canonical partition function identity

Question

The grand canonical partition function, Z, is given by $$ Z = \sum_{M=0}^\infty G(M)z^M =\frac{V_0(z)U_L(z)}{1−U(z)V (z)} \tag{1} $$ where G(M) is the canonical partition function of a chain of length M, z is the fugacity and $$ U(z) = \sum_{k=1}^\infty \frac{s^k}{k^c} z^k\quad V(z)=\sum_{k=1}^\infty \omega^k z^k $$ $V_0(z) = 1+V (z)$ and $U_L(z) = 1+U(z)$. The article I'm reading says that: "equation (1) can be verified by expanding the partition function as a series in U(z)V (z)". But .. in what way? Any suggestions please?