Complex analysis and Taylor development

Hey guys.
I need to prove this equation using Taylor development.
It's pretty obvious that if I'm only using the first part of the Taylor development (the part I marked in red) I can prove it, but my question is, is that enough? can I do something like that?

Thanks a lot.

If is a contour containing then:
where by Cauchy's theorem.
Therefore,

Thus, your equation seems to be wrong.