Can we obtain the Feynman diagrams using infinite series representation of a path integral? While evaluating quantum amplitude of a particle using path integral approach, we deal with infinite number of paths that can usually lead to a divergent infinite series. We can then also obtain a generalized Zeta function from the eigenvalues of the differential operator that appears in the action integral and then we regularize the path integral.
Can we somehow interpret this path integral in terms of infinite series and directly obtain the correlation functions and obtain the Feynman diagrams?
If this idea is possible, then how can we deal with the diverging infinite series that would appear?
 A: There is a link between Feynman diagrams and the path integral. Without loss of generality, we will specialise to the case of scalar fields for simplicity, such as $\phi^4$ theory described by,
$$\mathcal L  = \frac12 \partial_\mu \phi \partial^\mu \phi - \frac12 m^2 \phi^2 - \frac{\lambda}{4!}\phi^4.$$
The generating functional of the free theory is,
$$Z_0[J] = \int \mathcal D[\phi] \, e^{iS[\phi] + \int d^4x \, J(x)\phi(x)}  =\exp \left[ -\frac{i}{2} \int d^4x \, d^4y \, J(x)\Delta(x-y)J(y)\right] $$
where $\Delta(x-y)$ is the Feynman propagator. For the interacting theory, we have,
$$Z[J] = \frac{\exp \left[ -\frac{i\lambda}{4!} \int d^4z \left( \frac{1}{i}\frac{\delta}{\delta J(z)}\right)^4\right] Z_0[J]}{\exp \left[ -\frac{i\lambda}{4!} \int d^4z \left( \frac{1}{i}\frac{\delta}{\delta J(z)}\right)^4\right] Z_0[J] \bigg\rvert_{J=0}}.$$
$Z[J]$ can be expanded pictorially in terms of position-space Feynman diagrams; the first two terms, other than the constant, is a tadpole diagram, and the four-point vertex.

So, there is in fact an infinite series representation of the path integral, in terms of these diagrams. We can also link the path integral to scattering amplitudes, but you need to know we can represent time-ordered correlation functions as, for example, 
$$\langle \mathcal T \{ \phi(x_1)\phi(x_2)\} \rangle = -\frac{\delta^2}{\delta J(x_2) \delta(x_1)} Z[J]\bigg\rvert_{J=0}$$
that is, functional derivatives of the interacting theory's generating functional. As an example, we can now relate the scattering amplitude for $|p_1,p_2\rangle$ to $|p_3,\dots,p_n\rangle$ via the LSZ reduction formula:
$$\langle p_3, \dots, p_n | S | p_1,p_2 \rangle = i^n \int d^4x_1 e^{-ip_1 x_1}(\square_1 + m^2) \dots \prod_{i=1}^n \int d^4 x_i e^{ip_i x_i} (\square_i + m^2) \times$$
$$\times \langle \mathcal{T} \{\phi(x_1) \dots \phi(x_i)\}\rangle.$$
Thus to summarise:


*

*The path integral has a pictorial representation as a series of position-space Feynman diagrams.

*Correlation functions, since they can be expressed in terms of path integrals, also possess a Feynman diagram expansion.

*Through the LSZ reduction formula, scattering amplitudes and correlation functions of fields are linked.

