Equation 13.20 of Peskin & Schroeder I don't quite understand some skipped steps in the book An Introduction to Quantum Field Theory, by Peskin & Schroeder. Here is an example, I don't know why Taylor series would lead to this.

In Eq. (11.96), we showed that the effective action $\Gamma$ depends on the classical field $\phi_{\text{cl}}$ in such a way that the $n$th derivative of $\Gamma$ with respect to $\phi_{\text{cl}}$ gives the one-particle-irreducible $n$-point function of the field theory. Thus we can reconstruct $\Gamma$ from the 1PI functions by writing the Taylor series
$$\Gamma[\phi_{\text{cl}}] = i \sum_{2}^{\infty} \frac{1}{n!} \int \text{d}x_1 \cdots \text{d}x_n \phi_{\text{cl}}(x_1)\cdots\phi_{\text{cl}}(x_n) \Gamma^{(n)}(x_1, \ldots, x_n), \tag{13.20}$$
where the $\Gamma^{(n)}$ are the 1PI amplitudes.

 A: The object $\Gamma[\phi]$ (I'll drop the subscript to keep notation more simple) is a real number. We'd like to expand it in a Taylor series with respect to its dependence on the field variable $\phi$. If this was a regular function, we'd have
$$\Gamma[\phi] = \sum_{n=0}^{+\infty} \frac{1}{n!} \frac{\partial^n \Gamma}{\partial \phi^n} \phi^n.$$
However, $\phi$ has a dependence on spacetime. When we differentiate with respect to $\phi$, we must take that into consideration. This amounts to
$$\Gamma[\phi] = \sum_{n=0}^{+\infty} \frac{1}{n!} \int \frac{\delta^n \Gamma}{\delta\phi(x_1) \cdots \delta\phi(x_n)} \phi(x_1) \cdots \phi(x_n) \textrm{d}^d{x_1} \cdots \textrm{d}^d{x_n},$$
where each term is evaluated at $\phi = 0$.
To see this, let us pick an example. Consider $\Gamma[\phi] = \frac{m^2}{2} \int \phi^2 \textrm{d}^d{x}$. Using P&S's Eq. (9.31), we have
$$\frac{\delta \Gamma}{\delta \phi(x)} = m^2 \phi(x)$$
and
$$\frac{\delta^2 \Gamma}{\delta \phi(y) \delta \phi(x)} = m^2 \delta^{(d)}(x - y).$$
The remaining derivatives will vanish. If we plug in these expressions on the functional Taylor series I gave above, we have
$$\Gamma[\phi] = \frac{m^2}{2} \int 0 \ \textrm{d}^d{x} + m^2 \int 0 \cdot \phi(x) \textrm{d}^d{x} + \frac{1}{2} m^2 \int \phi(x)\phi(y) \delta^{(d)}(x - y) \textrm{d}^d{x}\textrm{d}^d{y} = \frac{m^2}{2} \int \phi^2 \textrm{d}^d{x},$$
which recovers exatcly the original expression.
To reach P&S's Eq. (13.20), we still have to employ Eq. (11.96). From the functional Taylor series expression, it leads to
$$\Gamma[\phi] = i \sum_{n=0}^{+\infty} \frac{1}{n!} \int \langle\hat{\phi}(x_1) \cdots \hat{\phi}(x_n) \rangle_{\text{1PI}} \phi(x_1) \cdots \phi(x_n) \textrm{d}^d{x_1} \cdots \textrm{d}^d{x_n},$$
where I use hats to denote the quantum fields. On Eq. (13.20), P&S change notation according to $\Gamma^{(n)}(x_1, \ldots, x_n) = \langle\hat{\phi}(x_1) \cdots \hat{\phi}(x_n) \rangle_{\text{1PI}}$, which leads to
$$\Gamma[\phi] = i \sum_{n=0}^{+\infty} \frac{1}{n!} \int \Gamma^{(n)}(x_1, \ldots, x_n) \phi(x_1) \cdots \phi(x_n) \textrm{d}^d{x_1} \cdots \textrm{d}^d{x_n},$$
as desired.
Functional derivatives occur not only on QFT, but also on Classical Field Theory. I suggest Section 11.3 of Nivaldo Lemos' Analytical Mechanics. It is an excellent, mathematically clear book with a nice account of Classical Field Theory. While it is not a Math book, it is much more careful than most Physics books and quite a good companion for QFT (and for life, if you ask me).
