Calculate $\int_{C} f(z)\ dz$ on the unit circle within a punctured disc.

Question

I have a function $f(z) = \sum_{n \geq -10} a_{n}z^{n}$, which converged absolutely on the punctured ($0$ removed) disc of radius 2. And I have a unit circled denoted by C. I want to find $\int_{C} f(z)\ dz$. Since we know the radius of convergence, we know that $f(z)$ is holomorphic in this disc except at $0$. And because of this hole in the domain, I cannot use Cauchy's Theorem. Then I think about using the extended Cauchy's Theorem, which says that if $C_{1}$ and $C_{2}$ are two simple closed curve in a region $R$. Then if $f(z)$ is holomorphic on $R$, then $\int_{C_{1}-C_{2}} f(z)\ dz = 0$. Then in this case, I let $$C_{1}(t)= e^{2 \pi i t}$$, where $t \in [0,1]$. And I let $$C_{2} (t)= ae^{2 \pi i t}$$, where $a \in (0,1), t \in [0,1]$. Then after I did the line integral, the result of $\int_{C_{1}} f(z)\ dz = 0$. I'm not sure if my idea is correct. Could anyone help me with this? Thank you!