# On Taylor Series Expansion and Complex Integrals

I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my solution because it is so abstract and I don't have anything to justify my answer with in the real world...

Say I want to solve $\oint_C \frac{\sin z}{z^2} dz$ where C is the unit circle. I do a Taylor series expansion and get:
$\frac{1}{z^2} [ \frac{1}{1!}z-\frac{1}{3!}z^3+ \frac{1}{5!}z^5-\ldots] = \frac{1}{z}-\frac{1}{3!}z+ \frac{1}{5!}z^3-\ldots$

My next step in reasoning is that I can use the Cauchy Integral Theorem on all the terms except $\frac{1}{z}$; in other words, these terms are all equivalent to 0.

I only need to use the Cauchy Integral Formula on the first term. This gives the solution of $2 \pi i$.

Is this a proper / appropriate rationale? Thanks.

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