I ask this question because similar questions don't address contour integrals and infinite series specifically, only evaluating a contour integral by converting it into an infinite series.
I'm trying to take the contour integral of an infinite series. My understanding is that under Fubini's theorem that under certain conditions, integrals can commute with each other, summations can commute with each other, and integrals can commute with summations.
However, the requirement (as I understand it) is that the summation must converge for the summation and integral to commute; but if I knew whether or not the summation converges, I would not need to integrate first.
Here's an example:
$$\oint_C \sum_{n=0}^ \infty a_{n} \cdot f_{n} \big(x\big) \stackrel{?}{=} \sum_{n=0}^ \infty \oint_C a_{n} \cdot f_{n} \big(x\big)$$
EDIT: I'd like to further explain my question and perhaps why it's confusing. I want to find an analytical solution to the area of a simply-connected space. I found that I could conveniently describe this space using parametric equations, and therefore could use Green's Theorem and integrate around the boundary to obtain the area. The parametric equations are:
$$x[\theta] = a \cdot Cos[\theta]$$
$$y[\theta] = \sum_{n=0}^ \infty \frac{b_{n}}{n} \cdot Sin[(2n+1) \cdot \theta] + \frac{c_{n}}{n} \cdot Cos[(2n+1) \cdot \theta ]$$
From there you can calculate $dx/d\theta$ and $dy/d\theta$ and from there $ A = \frac{1}{2} \oint_C x \cdot dy-y \cdot dx$.
However, summing the integral means summing the individual areas of each term's space, while integrating the sum means integrating the area from the sum of the contribution of each term to the coordinates at a given point on the contour.
These seem to be different operations to me and therefore I'm confused as to under what circumstances contour integral's and infinite series commute.
EDIT 2: I'm asking this question because I can do the integral on the right hand side (the sum of the contour integrals) in the example, but not the other (the contour integral of the sum), and need to know (1) how to test to see if both expressions are equivalent. The suggestions so far have been to integrate both sides and see if they are equivalent, but I can't do that, hence the need for Fubini's Theorem.
To elaborate on my comment, the Fubini-Tonelli theorem states the conditions for interchange of the limit operations, in particular for integrals. Summation is integration on a discrete space, so this is exactly the conditions for use of the theorem. As always, there are ways to break this if you do not have absolute convergence.
Beyond just the general case, Robert Israel correctly notes that the usual situation in complex analysis is that the series of functions converges absolutely and uniformly on compact subsets of some domain, in particular if the contour is a rectifiable curve in $\Bbb C$, the conditions are satisfied.