About series expansion of effective potential and its justification The books on quantum field theory often uses an expansion of the effective action $\Gamma[\phi_c]$ in terms of $\phi_{cl}$ and its derivatives given by $$
\Gamma[\phi_c]=\int d^4x[-V_{eff}(\phi_c)+Z(\phi_c)\frac{1}{2}(\partial_\mu\phi)^2+...]$$ However, I'm not sure how this formula is obtained, and what are the higher order terms in the series. Is it some generalized Taylor series? Can this formula be motivated or justified without its derivation? 
 A: This formula for the effective action is valid if (i) you are interested in a slowly varying field (i.e. in the low energy/momenta modes) and (ii) if the effective action not singular in this expansion.
If the expansion is well behaved, then you always have the right to do the expansion, which will obviously be valid only for fields varying slowly enough. This allows to obtain the inverse propagator, given by the second derivative of the effective action with respect to the fields (in momentum space, for a constant field $\bar \phi_c$, $G^{-1}(p;\bar\phi_c)=Z(\bar\phi_c) q^2-V''(\bar\phi_c)$).
However, note that it is not always possible to perform this expansion, in particular if the anomalous dimension is non-zero, as it happens usually for non-trivial RG fixed point. Indeed, in that case, at the minimum of the effective potential (assumed to be $\phi_c=0$ for simplicity), we have $G(q)\propto q^{-2+\eta}$, implying that $\partial_{q^2} \Gamma^{(2)}(q)|_{q=0}\to \infty$, which corresponds, in the derivative expansion to $Z\to \infty$.
Usually, this problem is cured, in the Wilsonian RG, by the infrared RG cut-off $k$. Because one has not integrated over the IR modes (the critical ones in phase transitions, giving rise to the divergence), one can safely expand the effective action in a gradient expansion. However, one finds that at a finite but small $k$ $Z_k\propto k^{-\eta}$, which obviously diverges as the RG scale is sent to zero. One can extract the anomalous dimension from this divergence. (In the QFT approach, one finds a two-loop correction to $Z$ of the form $-\eta \ln(q)$ which is also divergent at zero momentum.)
